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[[File:Angle.svg|alt=two line bent at a point|thumb|upright=1.25|A green angle formed by two red [[Ray (geometry)|rays]] on the [[Cartesian coordinate system]]]]
[[File:Angle.svg|alt=two line bent at a point|thumb|upright=1.25|A green angle formed by two red [[Ray (geometry)|rays]] on the [[Cartesian coordinate system]]]]


In [[Euclidean geometry]], an '''angle''' can refer to a number of concepts relating to the intersection of two straight [[Line (geometry)|lines]] at a point. Formally, an angle is a figure lying in a [[Euclidean plane|plane]] formed by two [[Ray (geometry)|rays]], called the ''[[Side (plane geometry)|sides]]'' of the angle, sharing a common endpoint, called the ''[[vertex (geometry)|vertex]]'' of the angle.<ref>{{Cite book |last=Hilbert |first=David |url=https://math.berkeley.edu/~wodzicki/160/Hilbert.pdf |title=The Foundations of Geometry |pages=9}}</ref><ref>{{harvnb|Sidorov|2001|ignore-err=yes}}</ref> More generally angles are also formed wherever two lines, rays or [[Line segment|line segments]] come together, such as at the corners of triangles and other polygons. An angle can be considered as the region of the plane bounded by the sides.<ref>{{Cite book |last=Evgrafov |first=M. A. |url=https://books.google.com/books?id=N8-wDwAAQBAJ&dq=angle+and+%2522angular+sector%2522+domain&pg=PA126 |title=Analytic Functions |date=2019-09-18 |publisher=Courier Dover Publications |isbn=978-0-486-84366-7 |language=en}}</ref><ref>{{Cite book |last=Papadopoulos |first=Athanase |url=https://books.google.com/books?id=f6yZeVMqhNEC&dq=angle+and+%2522angular+sector%2522+region&pg=PA12 |title=Strasbourg Master Class on Geometry |date=2012 |publisher=European Mathematical Society |isbn=978-3-03719-105-7 |language=en}}</ref>{{efn|An angular sector can be constructed by the combination of two rotated [[half-plane]]s, either their intersection or union (in the case of acute or obtuse angles, respectively).<ref>{{Cite book |last=D'Andrea |first=Francesco |url=https://books.google.com/books?id=BszREAAAQBAJ&dq=angle+and+%2522angular+sector%2522&pg=PA68 |title=A Guide to Penrose Tilings |date=2023-08-19 |publisher=Springer Nature |isbn=978-3-031-28428-1 |language=en}}</ref><ref>{{Cite book |last1=Bulboacǎ |first1=Teodor |url=https://books.google.com/books?id=r0miDwAAQBAJ&dq=angle+and+%2522angular+sector%2522+half-planes&pg=PT22 |title=Complex Analysis: Theory and Applications |last2=Joshi |first2=Santosh B. |last3=Goswami |first3=Pranay |date=2019-07-08 |publisher=Walter de Gruyter GmbH & Co KG |isbn=978-3-11-065803-3 |language=en}}</ref> It corresponds to a [[circular sector]] of infinite radius and a flat [[pencil of half-lines]].<ref>{{Cite book |last=Redei |first=L. |url=https://books.google.com/books?id=XMTSBQAAQBAJ&dq=half-pencil+of+lines&pg=PA45 |title=Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein |date=2014-07-15 |publisher=Elsevier |isbn=978-1-4832-8270-1 |language=en}}</ref>}} Angles can also be formed by the intersection of two planes or by two intersecting [[curve]]s, in which case the rays lying [[tangent]] to each curve at the point of intersection define the angle.
In [[Euclidean geometry]], an '''angle''' can refer to a number of concepts relating to the intersection of two straight [[Line (geometry)|lines]] at a [[Point (geometry)|point]]. Formally, an angle is a figure lying in a [[Euclidean plane|plane]] formed by two [[Ray (geometry)|rays]], called the ''[[Side (plane geometry)|sides]]'' of the angle, sharing a common endpoint, called the ''[[vertex (geometry)|vertex]]'' of the angle.<ref name=":0">{{Cite book |last=Hilbert |first=David |url=https://math.berkeley.edu/~wodzicki/160/Hilbert.pdf |title=The Foundations of Geometry |pages=9}}</ref><ref>{{harvnb|Sidorov|2001|ignore-err=yes}}</ref> More generally angles are also formed wherever two lines, rays or [[line segment]]s come together, such as at the corners of triangles and other polygons. An angle can be considered as the region of the plane bounded by the sides.<ref>{{Cite book |last=Evgrafov |first=M. A. |url=https://books.google.com/books?id=N8-wDwAAQBAJ&dq=angle+and+%2522angular+sector%2522+domain&pg=PA126 |title=Analytic Functions |date=2019-09-18 |publisher=Courier Dover Publications |isbn=978-0-486-84366-7 |language=en}}</ref><ref>{{Cite book |last=Papadopoulos |first=Athanase |url=https://books.google.com/books?id=f6yZeVMqhNEC&dq=angle+and+%2522angular+sector%2522+region&pg=PA12 |title=Strasbourg Master Class on Geometry |date=2012 |publisher=European Mathematical Society |isbn=978-3-03719-105-7 |language=en}}</ref>{{efn|An angular sector can be constructed by the combination of two rotated [[half-plane]]s, either their intersection or union (in the case of acute or obtuse angles, respectively).<ref>{{Cite book |last=D'Andrea |first=Francesco |url=https://books.google.com/books?id=BszREAAAQBAJ&dq=angle+and+%2522angular+sector%2522&pg=PA68 |title=A Guide to Penrose Tilings |date=2023-08-19 |publisher=Springer Nature |isbn=978-3-031-28428-1 |language=en}}</ref><ref>{{Cite book |last1=Bulboacǎ |first1=Teodor |url=https://books.google.com/books?id=r0miDwAAQBAJ&dq=angle+and+%2522angular+sector%2522+half-planes&pg=PT22 |title=Complex Analysis: Theory and Applications |last2=Joshi |first2=Santosh B. |last3=Goswami |first3=Pranay |date=2019-07-08 |publisher=Walter de Gruyter GmbH & Co KG |isbn=978-3-11-065803-3 |language=en}}</ref> It corresponds to a [[circular sector]] of infinite radius and a flat [[pencil of half-lines]].<ref>{{Cite book |last=Redei |first=L. |url=https://books.google.com/books?id=XMTSBQAAQBAJ&dq=half-pencil+of+lines&pg=PA45 |title=Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein |date=2014-07-15 |publisher=Elsevier |isbn=978-1-4832-8270-1 |language=en}}</ref>}} Angles can also be formed by the intersection of two planes or by two intersecting [[curve]]s, in which case the rays lying [[tangent]] to each curve at the point of intersection define the angle.


The term ''angle'' is also used for the size, [[magnitude (mathematics)|magnitude]] or [[Physical quantity|quantity]] of these types of geometric figures and in this context an angle consists of a number and unit of measurement. '''Angular measure''' or '''measure of angle''' are sometimes used to distinguish between the measurement and figure itself. The measurement of angles is intrinsically linked with circles and rotation. For an ordinary angle, this is often visualized or defined using the [[Arc (geometry)|arc]] of a [[circle]] centered at the vertex and lying between the sides.
The term ''angle'' is also used for the size, [[magnitude (mathematics)|magnitude]] or [[Physical quantity|quantity]] of these types of geometric figures and in this context an angle consists of a number and unit of measurement. '''Angular measure''' or '''measure of angle''' are sometimes used to distinguish between the measurement and figure itself. The measurement of angles is intrinsically linked with circles and rotation. For an ordinary angle, this is often visualized or defined using the [[Arc (geometry)|arc]] of a [[circle]] centered at the vertex and lying between the sides.


==Fundamentals==
== Fundamentals ==
An angle is a figure lying in a plane formed by two distinct rays (half-lines emanating indefinitely from an endpoint in one direction), which share a common endpoint. The rays are called the sides or arms of the angle, and the common endpoint is called the vertex. The sides divide the plane into two regions: the ''interior of the angle'' and the ''exterior of the angle.''
An angle is a figure lying in a plane formed by two distinct rays (half-lines emanating indefinitely from an endpoint in one direction), which share a common endpoint. The rays are called the sides or arms of the angle, and the common endpoint is called the vertex. The sides divide the plane into two regions: the ''interior of the angle'' and the ''exterior of the angle''.<ref name=":0" />


=== Notation ===
=== Notation ===
[[File:Angle diagram.svg|thumb|300x300px|<math>\angle BAC</math> is formed by rays <math>\vec{AB}</math> and '''<math>\vec{AC}</math>'''. '''<math>\theta</math>''' is the conventional measure of <math>\angle BAC</math> and '''<math>\beta</math>''' is an alternative measure.]]
[[File:Angle diagram.svg|thumb|<math>\angle \text{BAC}</math> is formed by rays <math>\vec{\text{AB}}</math> and <math>\vec{\text{AC}}</math>. <math>\theta</math> is the conventional measure of <math>\angle \text{BAC}</math> and <math>\beta</math> is an alternative measure.]]
In geometric figures and [[Expression (mathematics)|mathematical expressions]], it is common to use [[Greek letter]]s (<var>α</var>, <var>β</var>, <var>γ</var>, <var>θ</var>, <var>φ</var>,&nbsp;.&nbsp;.&nbsp;.&nbsp;) or lower case Roman letters (''a'',&nbsp;''b'',&nbsp;''c'',&nbsp;.&nbsp;.&nbsp;.&nbsp;) as [[Variable (mathematics)|variables]] denoting the size of an angle.{{sfn|Aboughantous|2010|p=18}} The Greek letter {{math|[[Pi (letter)|π]]}} is typically not used for this purpose to avoid confusion with the [[Pi|circle constant]].
An angle symbol (<math>\angle</math> or <math>\widehat{ \quad }</math>, read as "angle") together with one or three defining points is used to identify angles in geometric figures. For example, the angle with vertex A formed by the [[ray (geometry)|rays]] <math>\vec{\text{AB}}</math> and <math>\vec{\text{AC}}</math> is denoted as <math>\angle \text{A}</math> (using the vertex alone) or <math>\angle \text{BAC}</math> (with the vertex always named in the middle). The size or measure of the angle is denoted <math>m\angle \text{A}</math> or <math>m\angle \text{BAC}</math>.


An angle symbol (<math>\angle</math> or <math>\widehat{ \quad }</math>) with three defining points may also identify angles in geometric figures. For example, <math>\angle BAC</math> or <math>\widehat{BAC}</math> denotes the angle with vertex A formed by the [[ray (geometry)|rays]] AB and AC. Where there is no risk of confusion, the angle may sometimes be referred to by a single vertex alone (in this case, "angle A").
In geometric figures and [[Expression (mathematics)|mathematical expressions]], it is also common to use [[Greek letter]]s (''α'', ''β'', ''γ'', ''θ'', ''φ'',&nbsp;...) or lower case Roman letters (''a'',&nbsp;''b'',&nbsp;''c'',&nbsp;...) as [[Variable (mathematics)|variables]] to represent the size of an angle.{{sfn|Aboughantous|2010|p=18}}


Conventionally, angle size is measured "between" the sides through the interior of the angle and given as a [[Magnitude (mathematics)|magnitude]] or [[Scalar (physics)|scalar quantity]] without direction. At other times it might be a measure through the exterior of the angle or indicate a direction of measurement (see ''{{section link|#Signed angles}}'').
Conventionally, angle size is measured "between" the sides through the interior of the angle and given as a [[Magnitude (mathematics)|magnitude]] or [[Scalar (physics)|scalar quantity]]. At other times it might be measured through the exterior of the angle or given as a [[Sign (mathematics)#Signs per convention|signed number]] to indicate a direction of measurement.{{Citation needed|reason=The paragraph needs reliable sources|date=June 2025}}


=== Common angles and units of measurement ===
=== Units of measurement ===
{{Multiple image
Angles are measured in various units, the most common being the [[Degree (angle)|degree]] (denoted by the symbol '''°'''), [[radian]] (denoted by the symbol '''rad''') and [[Turn (angle)|turn]]. These units differ in the way they divide up a ''full angle'', an angle where one ray, initially congruent to the other, performs a compete rotation about the vertex to return back to its starting position.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Angle |url=https://mathworld.wolfram.com/Angle.html |access-date=2025-06-13 |website=mathworld.wolfram.com |language=en}}</ref>
|perrow = 2/3
|align=right
|direction=horizontal
|image1=Angle obtuse acute straight.svg
|width1=200
|caption1=Acute (<var>a</var>), obtuse (<var>b</var>), and straight (<var>c</var>) angles. All acute and obtuse angles are also oblique angles.


|image2=Zero angle.svg
Degrees and turns are defined directly with reference to a full angle, which measures 1 turn or 360°.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Degree |url=https://mathworld.wolfram.com/Degree.html |access-date=2025-06-14 |website=mathworld.wolfram.com |language=en}}</ref> A measure in turns gives an angle's size as a proportion of a full angle and a degree can be considered as a subdivision of a turn. Radians are not defined directly in relation to a full angle (see ''{{Section link|#Measuring angles}}''), but in such a way that its measure is 2{{math|π}}&nbsp;rad, approximately 6.28&nbsp;rad.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Radian |url=https://mathworld.wolfram.com/Radian.html |access-date=2025-06-14 |website=mathworld.wolfram.com |language=en}}</ref>
|width2=81
[[File:Angle types.svg|thumb|400x400px|Common types of angles]]
|caption2=Zero angle


|image3=Right angle.svg
=== Common angles ===
|width3=111
|caption3=[[Right angle]]
 
|image4=Reflex angle.svg
|width4=81
|caption4=Reflex angle
 
|image5=Full angle.svg
|width5=81
|caption5=Full angle
}}
Angles are measured in various units, the most common being the [[Degree (angle)|degree]] (denoted by the symbol '''°'''), [[radian]] (denoted by symbol '''rad''') and [[Turn (angle)|turn]]. These units differ in the way they divide up a ''full angle'', an angle where one ray, initially congruent to the other, performs a compete rotation about the vertex to return back to its starting position.
 
Degrees and turns are defined directly with reference to a full angle, which measures 1 turn or 360'''°'''. A measure in turns gives an angle's size as a proportion of a full angle and a degree can be considered as a subdivision of a turn. Radians are not defined directly in relation to a full angle (see {{Section link|2=Measuring angles|nopage=Y}}), but in such a way that its measure is <math>2\pi</math> rad, approximately 6.28 rad.
 
There is some common terminology for angles, whose conventional measure is always non-negative (see ''{{section link|#Signed angles}}''):
* An angle equal to 0° or not turned is called a ''zero angle''.{{sfn|Moser|1971|p=41}}
* An angle equal to 0° or not turned is called a ''zero angle''.{{sfn|Moser|1971|p=41}}
* An angle smaller than a right angle (less than 90°) is called an ''acute angle''{{sfn|Godfrey|Siddons|1919|p=9}} ("acute" meaning "[[Sharpness (visual)|sharp]]").
* An angle smaller than a right angle (less than 90°) is called an ''acute angle''.{{sfn|Godfrey|Siddons|1919|p=9}}
* An angle equal to {{sfrac|4}}&nbsp;[[Turn (angle)|turn]] (90° or {{sfrac|{{math|π}}|2}} [[radian]]s) is called a ''[[right angle]]''. Two lines that form a right angle are said to be ''[[normal (geometry)|normal]]'', ''[[orthogonality|orthogonal]]'', or ''[[perpendicular]]''.{{sfn|Moser|1971|p=71}}
* An angle equal to {{sfrac|4}}&nbsp;[[Turn (angle)|turn]] (90° or {{sfrac|{{math|π}}|2}}&nbsp;rad) is called a ''[[right angle]]''. Two lines that form a right angle are said to be ''[[normal (geometry)|normal]]'', ''[[orthogonality|orthogonal]]'', or ''[[perpendicular]]''.{{sfn|Moser|1971|p=71}}
* An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an ''obtuse angle''{{sfn|Godfrey|Siddons|1919|p=9}} ("obtuse" meaning "blunt").
* An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an ''obtuse angle''{{sfn|Godfrey|Siddons|1919|p=9}} ("obtuse" meaning "blunt").
* An angle equal to {{sfrac|2}}&nbsp;turn (180° or {{math|π}} radians) is called a ''straight angle''.{{sfn|Moser|1971|p=41}}
* An angle equal to {{sfrac|2}}&nbsp;turn (180° or {{math|π}}&nbsp;rad) is called a ''straight angle''.{{sfn|Moser|1971|p=41}}
* An angle larger than a straight angle but less than 1&nbsp;turn (between 180° and 360°) is called a ''reflex angle''.
* An angle larger than a straight angle but less than 1&nbsp;turn (between 180° and 360°) is called a ''reflex angle''.
* An angle equal to 1 turn (360° or 2{{math|π}} radians) is called a ''full angle'', ''complete angle'', ''round angle'' or ''perigon''.
* An angle equal to 1 turn (360° or 2{{math|π}}&nbsp;rad) is called a ''full angle'', ''complete angle'', ''round angle'' or ''perigon''.
* An angle that is not a multiple of a right angle is called an ''oblique angle''.
* An angle that is not a multiple of a right angle is called an ''oblique angle''.


Line 80: Line 57:
|style = "width:3em;" | {{nowrap|({{sfrac|1|2}}, 1) turn}}
|style = "width:3em;" | {{nowrap|({{sfrac|1|2}}, 1) turn}}
|style = "width:3em;" | {{nowrap|1 turn}}
|style = "width:3em;" | {{nowrap|1 turn}}
|-
| style="background:#f2f2f2; text-align:center;" | [[Degree (angle)|degree]]&nbsp;&nbsp;
| style="width:3em;" | 0°
| style="width:3em;" | (0,&nbsp;90)°
| style="width:3em;" | 90°
| style="width:3em;" | (90,&nbsp;180)°
| style="width:3em;" | 180°
| style="width:3em;" | (180,&nbsp;360)°
| style="width:3em;" | 360°
|-
|-
|style = "background:#f2f2f2; text-align:center;" | [[radian]]
|style = "background:#f2f2f2; text-align:center;" | [[radian]]
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| {{nowrap|(''{{pi}}'', 2''{{pi}}'') rad}}
| {{nowrap|(''{{pi}}'', 2''{{pi}}'') rad}}
| {{nowrap|2''{{pi}}'' rad}}
| {{nowrap|2''{{pi}}'' rad}}
|-
|style = "background:#f2f2f2; text-align:center;" | [[Degree (angle)|degree]]&nbsp;&nbsp;
|style = "width:3em;" | 0°
|style = "width:3em;" | (0,&nbsp;90)°
|style = "width:3em;" | 90°
|style = "width:3em;" | (90,&nbsp;180)°
|style = "width:3em;" | 180°
|style = "width:3em;" | (180,&nbsp;360)°
|style = "width:3em;" | 360°
|-
|style = "background:#f2f2f2; text-align:center;" | [[Gradian|gon]]&nbsp;&nbsp;
|style = "width:3em;" | 0<sup>g</sup>
|style = "width:3em;" | (0,&nbsp;100)<sup>g</sup>
|style = "width:3em;" | 100<sup>g</sup>
|style = "width:3em;" | (100,&nbsp;200)<sup>g</sup>
|style = "width:3em;" | 200<sup>g</sup>
|style = "width:3em;" | (200,&nbsp;400)<sup>g</sup>
|style = "width:3em;" | 400<sup>g</sup>
|-
|-
|}
|}


==Types{{anchor|Types of angles}}==
=== Addition and subtraction ===
[[File:Angle addition.svg|thumb|The angle addition postulate defines addition and subtraction of angles: {{nowrap|1=''θ'' + ''α'' = ''φ''}}; {{nowrap|1=''φ'' − ''α'' = ''θ''}}.]]
The '''angle addition postulate''' states that if D is a point lying in the interior of <math>\angle \text{BAC}</math> then:<ref>{{cite book |last=Moise |first=Edwin, E |url=https://www.ime.usp.br/~toscano/disc/2021/Moise.pdf |title=Elementary geometry from an advanced standpoint |publisher=Addison-Wesley Publishing Company |year=1990 |edition=3rd |pages=96}}</ref> <math display="block"> m\angle \text{BAC} = m\angle \text{BAD} + m\angle \text{DAC}.</math> This relationship ''defines'' what it means add any two angles: their vertices are placed together while sharing a side to create a new larger angle. The measure of the new larger angle is the sum of the measures of the two angles. Subtraction follows from rearrangement of the formula.{{Citation needed|reason=The paragraph needs reliable sources|date=June 2025}}
 
== Types ==
{{Redirect|Oblique angle|the cinematographic technique|Dutch angle}}
{{Redirect|Oblique angle|the cinematographic technique|Dutch angle}}


===Vertical and {{vanchor|adjacent}} angle pairs===
=== Adjacent and vertical angles ===
[[File:Vertical Angles.svg|thumb|150px|right|Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles. [[Hatch_mark#Congruency_notation|Hatch marks]] are used here to show angle equality.]]
[[File:Adjacentangles.svg|right|thumb|225x225px|Angles A and B are adjacent.]]
{{redirect-distinguish|Vertical angle|Zenith angle}}
{{redirect-distinguish|Vertical angle|Zenith angle}}


When two straight lines intersect at a point, four angles are formed. Pairwise, these angles are named according to their location relative to each other.
''Adjacent angles'' (abbreviated ''adj. ∠s''), are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively called ''complementary'', ''supplementary'', and ''explementary'' angles (see ''{{section link|#Combining angle pairs}}'' below).
{{bulleted list
[[File:Vertical Angles.svg|thumb|150px|Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles. [[Hatch_mark#Congruency_notation|Hatch marks]] are used here to show angle equality.]]''Vertical angles'' are formed when two straight lines intersect at a point producing four angles. A pair of angles opposite each other are called ''vertical angles'', ''opposite angles'' or ''vertically opposite angles'' (abbreviated ''vert. opp. ∠s''),<ref name="tb">{{harvnb|Wong|Wong|2009|pp=161–163}}</ref> where "vertical" refers to the sharing of a vertex, rather than an up-down orientation. The ''vertical angle theorem'' states that vertical angles are always congruent or equal to each other.{{Citation needed|reason=This sentence needs reliable sources|date=June 2025}}
| A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called ''vertical angles'' or ''opposite angles'' or ''vertically opposite angles''. They are abbreviated as ''vert. opp. ∠s''.<ref name="tb">{{harvnb|Wong|Wong|2009|pp=161–163}}</ref>
A [[Transversal (geometry)|transversal]] is a line that intersects a pair of (often parallel) lines and is associated with ''exterior angles'', ''interior angles'', ''alternate exterior angles'', ''alternate interior angles'', ''corresponding angles'', and ''consecutive interior angles''.{{sfn|Jacobs|1974|p=255}}
{{pb}}
 
The equality of vertically opposite angles is called the ''vertical angle theorem''. [[Eudemus of Rhodes]] attributed the proof to [[Thales|Thales of Miletus]].<ref>{{cite book|author=Euclid|author-link=Euclid|title=The Elements|title-link=Euclid's Elements}} Proposition I:13.</ref>{{sfn|Shute| Shirk|Porter|1960|pp=25–27}} The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note,{{sfn|Shute| Shirk|Porter|1960|pp=25–27}} when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as:
=== Combining angle pairs ===
* All straight angles are equal.
{{anchor|Angle addition postulate}}
* Equals added to equals are equal.
 
* Equals subtracted from equals are equal.
When summing two angles that are either adjacent or separated in space, three cases are of particular importance.{{Citation needed|reason=This sentence needs reliable sources|date=June 2025}}
 
==== Complementary angles ====
[[File:Complement angle.svg|thumb|150px|Angles a and b are ''complementary'' angles]]
''Complementary angles'' are angle pairs whose measures sum to a right angle ({{sfrac|4}} turn, 90°, or {{sfrac|{{math|π}}|2}} rad).<ref>{{Cite web |title=Complementary Angles |url=https://www.mathsisfun.com/geometry/complementary-angles.html |access-date=2020-08-17 |website=www.mathsisfun.com}}</ref> If the two complementary angles are adjacent, their non-shared sides form a right angle. In a [[Right triangle|right-angle triangle]] the two acute angles are complementary as the sum of the internal angles of a [[triangle]] is 180°.


When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle ''A'' equals ''x'', the measure of angle ''C'' would be {{nowrap|180° − ''x''}}. Similarly, the measure of angle ''D'' would be {{nowrap|180° − ''x''}}. Both angle ''C'' and angle ''D'' have measures equal to {{nowrap|180° − ''x''}} and are congruent. Since angle ''B'' is supplementary to both angles ''C'' and ''D'', either of these angle measures may be used to determine the measure of Angle ''B''. Using the measure of either angle ''C'' or angle ''D'', we find the measure of angle ''B'' to be {{nowrap|1=180° − (180° − ''x'') = 180° − 180° + ''x'' = ''x''}}. Therefore, both angle ''A'' and angle ''B'' have measures equal to ''x'' and are equal in measure.
The difference between an angle and a right angle is termed the ''complement'' of the angle<ref name="Chisholm 19112">{{harvnb|Chisholm|1911}}</ref> which is from the Latin ''complementum'' and associated verb ''complere'', meaning "to fill up". An acute angle is "filled up" by its complement to form a right angle.{{Citation needed|reason=The claims after the source need reliable sources|date=June 2025}}


[[File:Adjacentangles.svg|right|thumb|225px|Angles ''A'' and ''B'' are adjacent.]]
| ''Adjacent angles'', often abbreviated as ''adj. ∠s'', are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively called ''complementary'', ''supplementary'', and ''explementary'' angles (see ''{{section link|#Combining angle pairs}}'' below).
}}
A [[Transversal (geometry)|transversal]] is a line that intersects a pair of (often parallel) lines and is associated with ''exterior angles'', ''interior angles'', ''alternate exterior angles'', ''alternate interior angles'', ''corresponding angles'', and ''consecutive interior angles''.{{sfn|Jacobs|1974|p=255}}


===Combining angle pairs===
{{anchor|Angle addition postulate}}The '''angle addition postulate''' states that if B is in the interior of angle AOC, then


<math display="block"> m\angle \mathrm{AOC} = m\angle \mathrm{AOB} + m\angle \mathrm{BOC} </math>
==== Supplementary angles ====
[[File:Angle obtuse acute straight.svg|thumb|300x300px|Angles <var>a</var> and <var>b</var> are ''supplementary'' angles]]
Two angles that sum to a straight angle ({{sfrac|2}} turn, 180°, or {{math|π}} rad) are called ''supplementary angles''.<ref>{{Cite web |title=Supplementary Angles |url=https://www.mathsisfun.com/geometry/supplementary-angles.html |access-date=2020-08-17 |website=www.mathsisfun.com}}</ref> If the two supplementary angles are [[Angle#adjacent|adjacent]], their non-shared sides form a straight angle or [[Line (geometry)|straight line]] and are called a ''linear pair of angles''.{{sfn|Jacobs|1974|p=97}}  The difference between an angle and a straight angle is termed the ''supplement'' of the angle.<ref>{{Cite book |last=Rhoad |first=Richard |url=http://archive.org/details/geometryforenjoy00geor |title=Geometry for enjoyment and challenge |last2=Milauskas |first2=George |last3=Whipple |first3=Robert |last4=McDougal Littell |date=1991 |publisher=Evanston, Ill. : McDougal, Littell |others=Internet Archive |isbn=978-0-86609-965-3 |page=67}}</ref>


I.e., the measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC.
Examples of non-adjacent complementary angles include the consecutive angles of a [[parallelogram]] and opposite angles of a [[cyclic quadrilateral]]. For a circle with center O, and [[Tangent lines to circles|tangent lines]] from an exterior point P touching the circle at points T and Q, the resulting angles ∠TPQ and ∠TOQ are supplementary.


Three special angle pairs involve the summation of angles:
==== Explementary angles ====
{{anchor|complementary angle}}
[[File:Conjugate_Angles.svg|thumb|Angles AOB and COD are ''explementary'' ''or conjugate'' angles]]
[[File:Complement angle.svg|thumb|150px|The ''complementary'' angles <var>a</var> and <var>b</var> (<var>b</var> is the ''complement'' of <var>a</var>, and <var>a</var> is the complement of <var>b</var>.)]]
Two angles that sum to a full angle (1 turn, 360°, or 2{{math|π}} radians) are called ''explementary angles'' or ''conjugate angles''.<ref>{{cite book |last=Willis |first=Clarence Addison |url=https://archive.org/details/planegeometryexp00willrich/page/8/ |title=Plane Geometry |publisher=Blakiston's Son |year=1922 |page=8}}</ref> The difference between an angle and a full angle is termed the ''explement'' or ''conjugate'' of the angle.{{Citation needed|reason=This sentence needs reliable sources|date=June 2025}}
{{bulleted list
| ''Complementary angles'' are angle pairs whose measures sum to one right angle ({{sfrac|4}} turn, 90°, or {{sfrac|{{math|π}}|2}} radians).<ref>{{Cite web|title=Complementary Angles|url=https://www.mathsisfun.com/geometry/complementary-angles.html|access-date=2020-08-17 | website=www.mathsisfun.com}}</ref> If the two complementary angles are adjacent, their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary because the sum of internal angles of a [[triangle]] is 180 degrees, and the right angle accounts for 90 degrees.
{{pb}}
The adjective complementary is from the Latin ''complementum'', associated with the verb ''complere'', "to fill up". An acute angle is "filled up" by its complement to form a right angle.
{{pb}}
The difference between an angle and a right angle is termed the ''complement'' of the angle.<ref name="Chisholm 1911">{{harvnb|Chisholm|1911}}</ref>
{{pb}}
If angles ''A'' and ''B'' are complementary, the following relationships hold: <math display="block">
\begin{align}
& \sin^2A + \sin^2B = 1 & & \cos^2A + \cos^2B = 1 \\[3pt]
& \tan A = \cot B  & & \sec A = \csc B
\end{align}</math>
{{pb}}
(The [[tangent]] of an angle equals the [[cotangent]] of its complement, and its secant equals the [[cosecant]] of its complement.)
{{pb}}
The [[prefix]] "[[co (function prefix)|co-]]" in the names of some trigonometric ratios refers to the word "complementary".
{{clear|right}}
[[File:Angle obtuse acute straight.svg|thumb|right|300px|The angles <var>a</var> and <var>b</var> are ''supplementary'' angles.]]
| {{anchor|Linear pair of angles|Supplementary angle}}Two angles that sum to a straight angle ({{sfrac|2}} turn, 180°, or {{math|π}} radians) are called ''supplementary angles''.<ref>{{Cite web|title=Supplementary Angles|url=https://www.mathsisfun.com/geometry/supplementary-angles.html|access-date=2020-08-17 | website=www.mathsisfun.com}}</ref>
{{pb}}
If the two supplementary angles are [[#adjacent|adjacent]] (i.e., have a common [[vertex (geometry)|vertex]] and share just one side), their non-shared sides form a [[line (geometry)|straight line]]. Such angles are called a ''linear pair of angles''.{{sfn|Jacobs|1974|p=97}} However, supplementary angles do not have to be on the same line and can be separated in space. For example, adjacent angles of a [[parallelogram]] are supplementary, and opposite angles of a [[cyclic quadrilateral]] (one whose vertices all fall on a single circle) are supplementary.
{{pb}}
If a point P is exterior to a circle with center O, and if the [[tangent lines to circles|tangent lines]] from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.
{{pb}}
The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.
{{pb}}
In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third because the sum of the internal angles of a triangle is a straight angle.
{{clear|right}}
{{anchor|explementary angle}}
[[File:Conjugate Angles.svg|thumb|Angles AOB and COD are conjugate as they form a complete angle. Considering magnitudes, 45° + 315° = 360°.]]
| Two angles that sum to a complete angle (1 turn, 360°, or 2{{math|π}} radians) are called ''explementary angles'' or ''conjugate angles''.<ref>{{cite book |last=Willis |first=Clarence Addison |year=1922 |publisher=Blakiston's Son |title=Plane Geometry |page=8 |url=https://archive.org/details/planegeometryexp00willrich/page/8/ }}</ref>
{{pb}}
The difference between an angle and a complete angle is termed the ''explement'' of the angle or ''conjugate'' of an angle.
{{clear|right}}
}}


===Polygon-related angles===
=== Polygon-related angles ===
[[File:ExternalAngles.svg|thumb|300px|right|Internal and external angles]]
[[File:ExternalAngles.svg|thumb|300px|right|Internal and external angles]]
* An angle that is part of a [[simple polygon]] is called an ''[[interior angle]]'' if it lies on the inside of that simple polygon. A simple [[concave polygon]] has at least one interior angle, that is, a reflex angle. {{pb}} <!-- --> In [[Euclidean geometry]], the measures of the interior angles of a [[triangle]] add up to {{math|π}} radians, 180°, or {{sfrac|2}} turn; the measures of the interior angles of a simple [[convex polygon|convex]] [[quadrilateral]] add up to 2{{math|π}} radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex [[polygon]] with ''n'' sides add up to (''n''&nbsp;−&nbsp;2){{math|π}}&nbsp;radians, or (''n''&nbsp;−&nbsp;2)180&nbsp;degrees, (''n''&nbsp;−&nbsp;2)2 right angles, or (''n''&nbsp;−&nbsp;2){{sfrac|1|2}}&nbsp;turn.
* An angle that is part of a [[simple polygon]] is called an ''[[interior angle]]'' if it lies on the inside of that simple polygon. A simple [[concave polygon]] has at least one interior angle, that is, a reflex angle. {{pb}} <!-- --> In [[Euclidean geometry]], the measures of the interior angles of a [[triangle]] add up to {{math|π}} radians, 180°, or {{sfrac|2}} turn; the measures of the interior angles of a simple [[convex polygon|convex]] [[quadrilateral]] add up to 2{{math|π}} radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex [[polygon]] with ''n'' sides add up to (''n''&nbsp;−&nbsp;2){{math|π}}&nbsp;radians, or (''n''&nbsp;−&nbsp;2)180&nbsp;degrees, (''n''&nbsp;−&nbsp;2)2 right angles, or (''n''&nbsp;−&nbsp;2){{sfrac|1|2}}&nbsp;turn.
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* In a triangle, three intersection points, each of an external angle bisector with the opposite [[extended side]], are [[collinearity|collinear]].<ref name=Johnson/>{{rp|p=149}}
* In a triangle, three intersection points, each of an external angle bisector with the opposite [[extended side]], are [[collinearity|collinear]].<ref name=Johnson/>{{rp|p=149}}
* In a triangle, three intersection points, two between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended are collinear.<ref name=Johnson/>{{rp|p=149}}
* In a triangle, three intersection points, two between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended are collinear.<ref name=Johnson/>{{rp|p=149}}
* Some authors use the name ''exterior angle'' of a simple polygon to mean the ''explement exterior angle'' (''not'' supplement!) of the interior angle.<ref>{{citation|editor=D. Zwillinger|title=CRC Standard Mathematical Tables and Formulae|place=Boca Raton, FL|publisher=CRC Press | year=1995 | page= 270}} as cited in {{MathWorld |urlname=ExteriorAngle |title=Exterior Angle}}</ref> This conflicts with the above usage.
* Some authors use the name ''exterior angle'' of a simple polygon to mean the ''explement exterior angle'' (''not'' supplement!) of the interior angle.<ref>{{citation |editor=D. Zwillinger |title=CRC Standard Mathematical Tables and Formulae |place=Boca Raton, FL |publisher=CRC Press | year=1995 | page= 270}} as cited in {{MathWorld |urlname=ExteriorAngle |title=Exterior Angle}}</ref> This conflicts with the above usage.


===Plane-related angles===
=== Plane-related angles ===
* The angle between two [[Plane (mathematics)|planes]] (such as two adjacent faces of a [[polyhedron]]) is called a ''[[dihedral angle]]''.<ref name="Chisholm 1911"/> It may be defined as the acute angle between two lines [[Normal (geometry)|normal]] to the planes.
* The angle between two [[Plane (mathematics)|planes]] (such as two adjacent faces of a [[polyhedron]]) is called a ''[[dihedral angle]]''.<ref name="Chisholm 19112" /> It may be defined as the acute angle between two lines [[Normal (geometry)|normal]] to the planes.
* The angle between a plane and an intersecting straight line is complementary to the angle between the intersecting line and the [[normal (geometry)|normal]] to the plane.
* The angle between a plane and an intersecting straight line is complementary to the angle between the intersecting line and the [[normal (geometry)|normal]] to the plane.


==Measuring angles{{anchor|Measurement}}==<!-- linked from [[Degree (angle)]] -->
== Measuring angles <span class="anchor" id="Measurement"></span> ==
<!-- linked from [[Degree (angle)]] -->
{{see also|Angle measuring instrument}}
{{see also|Angle measuring instrument}}
[[File:Basic angle in circle.svg|thumb|300x300px|The angle size <math>\theta</math> can be measured as s/r radians or s/C turns]]
[[File:Basic angle in circle.svg|thumb|300x300px|The angle size <math>\theta</math> can be measured as s/r radians or s/C turns]]
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The ratio of the length <var>s</var> by the radius <var>r</var> is the number of [[radian]]s in the angle, while the ratio of length <var>s</var> by the circumference <var>C</var> is the number of [[Turn (angle)|turns]]:<ref name="SIBrochure9thEd">{{citation |author=International Bureau of Weights and Measures |title=The International System of Units (SI) |date=20 May 2019 |url=https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf |archive-url=https://web.archive.org/web/20211018184555/https://www.bipm.org/documents/20126/41483022/SI-Brochure-9.pdf/fcf090b2-04e6-88cc-1149-c3e029ad8232 |archive-date=18 October 2021 |url-status=live |edition=9th |isbn=978-92-822-2272-0 |author-link=New SI}}</ref>
The ratio of the length <var>s</var> by the radius <var>r</var> is the number of [[radian]]s in the angle, while the ratio of length <var>s</var> by the circumference <var>C</var> is the number of [[Turn (angle)|turns]]:<ref name="SIBrochure9thEd">{{citation |author=International Bureau of Weights and Measures |title=The International System of Units (SI) |date=20 May 2019 |url=https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf |archive-url=https://web.archive.org/web/20211018184555/https://www.bipm.org/documents/20126/41483022/SI-Brochure-9.pdf/fcf090b2-04e6-88cc-1149-c3e029ad8232 |archive-date=18 October 2021 |url-status=live |edition=9th |isbn=978-92-822-2272-0 |author-link=New SI}}</ref>
<math display="block"> \theta = \frac{s}{r} \, \mathrm{rad}. </math><math display="block"> \theta = \frac{s}{ C} \ = \frac{s}{2\pi r} \, \mathrm{turns} </math>
<math display="block">\theta_\mathrm{rad} = \frac{s}{r} \, \mathrm{rad} \qquad \qquad \theta_{\mathrm{turn}} = \frac{s}{ C} \ = \frac{s}{2\pi r} \, \mathrm{turns}</math>


[[File:Angle measure.svg|right|thumb|The measure of angle {{math|''θ''}} is {{nowrap|{{sfrac|''s''|''r''}} radians}}.]]
[[File:Angle measure.svg|right|thumb|The measure of angle {{math|''θ''}} is {{nowrap|{{sfrac|''s''|''r''}} radians}}.]]


The value of {{math|''θ''}} thus defined is independent of the size of the circle: if the length of the radius is changed, then both the circumference and the arc length change in the same proportion, so the ratios <math>\frac{s}{r}</math>and <math>\frac{s}{C}</math> are unaltered.{{refn|group="nb"|This approach requires, however, an additional proof that the measure of the angle does not change with changing radius {{math|''r''}}, in addition to the issue of "measurement units chosen". A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See Radoslav M. Dimitrić, for instance.<ref name="Dimitric_2012"/>}}
The value of {{math|''θ''}} thus defined is independent of the size of the circle: if the length of the radius is changed, then both the circumference and the arc length change in the same proportion, so the ratios {{sfrac|''s''|''r''}} and {{sfrac|''s''|''C''}} are unaltered.{{refn|group="nb"|This approach requires, however, an additional proof that the measure of the angle does not change with changing radius {{math|''r''}}, in addition to the issue of "measurement units chosen". A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See Radoslav M. Dimitrić, for instance.<ref name="Dimitric_2012"/>}}


Angles of the same size are said to be ''equal'' ''congruent'' or ''equal in measure''.
Angles of the same size are said to be ''equal'' ''congruent'' or ''equal in measure''.


=== Units ===
=== Units ===
In addition to the radian and turn, other angular units exist, typically based on subdivisions of the turn, including the [[degree (angle)|degree]] ( ° ) and the [[gradian]] (grad), though many others have been used throughout [[History of Mathematics|history]].<ref>{{Cite web |title=angular unit |url=https://www.thefreedictionary.com/angular+unit |access-date=2020-08-31 |website=TheFreeDictionary.com}}</ref>  
In addition to the radian and turn, other angular units exist, typically based on subdivisions of the turn, including the [[degree (angle)|degree]] (°) and the [[gradian]] (grad), though many others have been used throughout [[History of Mathematics|history]].<ref>{{Cite web |title=angular unit |url=https://www.thefreedictionary.com/angular+unit |access-date=2020-08-31 |website=TheFreeDictionary.com}}</ref>


Conversion between units may be obtained by multiplying the anglular measure in one unit by a conversion constant of the form <math>\frac{k_a}{k_b}</math> where <math>{k_a}</math> and <math>{k_b}</math> are the measures of a complete turn expressed in units a and b. For example, {{nowrap|1= ''k'' = 360°}} for [[degree (angle)|degrees]] or 400 grad for [[gradian]]s):<math display="block"> \theta_\deg = \frac{360}{2\pi} \cdot \theta </math>The following table lists some units used to represent angles.
Conversion between units may be obtained by multiplying the angular measure in one unit by a conversion constant of the form <math>\tfrac{k_a}{k_b}</math> where <math>{k_a}</math> and <math>{k_b}</math> are the measures of a complete turn in units ''a'' and ''b''. For example, to convert an angle of <math> \tfrac{\pi}{2} </math>radians to degrees:
<math display="block">\theta_\deg = \frac{k_\deg}{k_\mathrm{rad}} \cdot \theta_\mathrm{rad} = \frac{360^{\circ}}{2\pi \,\mathrm{rad}} \cdot \frac{\pi}{2} \, \mathrm{rad} = 90^{\circ}</math>


The following table lists some units used to represent angles.
{|class = "wikitable"
{|class = "wikitable"
!Name !!Number in one turn!!In degrees !!Description
!Name (symbol) !! Number in one turn !! 1 unit in degrees !! Description
|-
|-
|[[radian]]||{{math|2''π''}}||≈57°17′45″||The ''radian'' is determined by the circumference of a circle that is equal in length to the radius of the circle (''n''&nbsp;=&nbsp;2{{pi}}&nbsp;=&nbsp;6.283...). It is the angle subtended by an arc of a circle that has the same length as the circle's radius. The symbol for radian is ''rad''. One turn is 2{{math|π}}&nbsp;radians, and one radian is {{sfrac|180°|{{pi}}}}, or about 57.2958 degrees. Often, particularly in mathematical texts, one radian is assumed to equal one, resulting in the unit ''rad'' being omitted. The radian is used in virtually all mathematical work beyond simple, practical geometry due, for example, to the pleasing and "natural" properties that the [[trigonometric function]]s display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the [[SI]].
| [[turn (geometry)|turn]] || 1 || 360° || The ''turn'' is the angle subtended by the circumference of a circle at its centre. A turn is equal to 2{{pi}} or [[Tau (mathematics)|{{tau}}]] radians.
|-
|-
|[[degree (angle)|degree]] ||360 ||1°|| The ''degree'', denoted by a small superscript circle (°), is 1/360 of a turn, so one ''turn'' is 360°. One advantage of this old [[sexagesimal]] subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g., 3.5° for three and a half degrees), but the [[Minute and second of arc|"minute" and "second"]] sexagesimal subunits of the "degree–minute–second" system (discussed next) are also in use, especially for [[Geographic coordinate system|geographical coordinates]] and in [[astronomy]] and [[ballistics]] (''n''&nbsp;=&nbsp;360)
| [[degree (angle)|degree]] (&nbsp;°&nbsp;)
| 360 || 1°|| The ''degree'' is a [[sexagesimal]] subunit of the sextant, making one ''turn'' equal to 360°.
|-
|-
| [[arcminute]]||21,600 ||0°1′|| The ''minute of arc'' (or ''MOA'', ''arcminute'', or just ''minute'') is {{sfrac|60}} of a degree = {{sfrac|21,600}} turn. It is denoted by a single prime (&nbsp;′&nbsp;). For example, 3°&nbsp;30′ is equal to 3&nbsp;×&nbsp;60&nbsp;+&nbsp;30&nbsp;=&nbsp;210 minutes or 3&nbsp;+&nbsp;{{sfrac|30|60}} = 3.5 degrees. A mixed format with decimal fractions is sometimes used, e.g., 3°&nbsp;5.72′ = 3&nbsp;+&nbsp;{{sfrac|5.72|60}} degrees. A [[nautical mile]] was historically defined as an arcminute along a [[great circle]] of the Earth. (''n''&nbsp;=&nbsp;21,600).
| [[radian]] (rad)
| {{math|2''π''}} || 57.2957...° || The ''radian'' is the angle subtended by an arc of a circle that has the same length as the circle's radius.
|-
|-
| [[arcsecond]]||1,296,000 ||0°0′1″||The ''second of arc'' (or ''arcsecond'', or just ''second'') is {{sfrac|60}} of a minute of arc and {{sfrac|3600}} of a degree (''n''&nbsp;=&nbsp;1,296,000). It is denoted by a double prime (&nbsp;″&nbsp;). For example, 3°&nbsp;7′&nbsp;30″ is equal to 3 + {{sfrac|7|60}} + {{sfrac|30|3600}} degrees, or 3.125&nbsp;degrees. The arcsecond is the angle used to measure a [[parsec]]
| [[grad (angle)|grad]] (gon)
| 400 || 0.9° || The ''grad'', also called ''grade'', ''[[gradian]]'', or ''gon'', is defined as {{sfrac|1|100}} of a right angle. The grad is used mostly in [[triangulation (surveying)|triangulation]] and continental [[surveying]].
|-
|-
| [[grad (angle)|grad]]||400 ||0°54′ || The ''grad'', also called ''grade'', ''[[gradian]]'', or ''gon''. It is a decimal subunit of the quadrant. A right angle is 100 grads. A [[kilometre]] was historically defined as a [[centi]]-grad of arc along a [[meridian (geography)|meridian]] of the Earth, so the kilometer is the decimal analog to the [[sexagesimal]] [[nautical mile]] (''n''&nbsp;=&nbsp;400). The grad is used mostly in [[triangulation (surveying)|triangulation]] and continental [[surveying]].
| [[arcminute]] (&nbsp;′&nbsp;)
|21,600 || {{sfrac|1|60}}° || The ''minute of arc'' (or ''arcminute'', or just ''minute'') is a sexagesimal subunit of a degree.
|-
|-
|[[turn (geometry)|turn]]||1||360° || The ''turn'' is the angle subtended by the circumference of a circle at its centre. A turn is equal to 2{{pi}} or [[Turn_(angle)#Proposals_for_a_single_letter_to_represent_2π|{{tau}} (tau)]] radians.
| [[arcsecond]] (&nbsp;″&nbsp;)
|1,296,000 || {{sfrac|1|3600}}° ||The ''second of arc'' (or ''arcsecond'', or just ''second'') is a sexagesimal subunit of a minute of arc.
|-
|-
| [[hour angle]] || 24 || 15° || The astronomical ''hour angle'' is {{sfrac|24}}&nbsp;turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called ''minute of time'' and ''second of time''. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1&nbsp;hour = 15° = {{sfrac|{{pi}}|12}}&nbsp;rad = {{sfrac|6}}&nbsp;quad = {{sfrac|24}}&nbsp;turn = {{sfrac|16|2|3}}&nbsp;grad.
| [[milliradian]] (mrad)
| {{math|2000''π''}} || ~0.0573° || The milliradian is a thousandth of a radian. For artillery and navigation a unit is used, often called a 'mil', which are ''approximately'' equal to a milliradian. One turn is exactly 6000, 6300, or 6400 mils, depending on which definition is used.
|-
|-
| [[Points of the compass|(compass) point]] || 32 || 11°15′ || The ''point'' or ''wind'', used in [[navigation]], is {{sfrac|32}} of a turn. 1&nbsp;point = {{sfrac|8}} of a right angle = 11.25° = 12.5&nbsp;grad. Each point is subdivided into four quarter points, so one turn equals 128.
| [[Points of the compass|(compass) point]] || 32 || 11.25° || The ''point'' or ''wind'', used in [[navigation]], divides the compass (one turn) into 32 points or compass directions.
|-
|-
| [[milliradian]] || {{math|2000''π''}} || ≈0.057° || The true milliradian is defined as a thousandth of a radian, which means that a rotation of one [[Turn (geometry)|turn]] would equal exactly 2000π&nbsp;mrad (or approximately 6283.185&nbsp;mrad). Almost all [[Telescopic sight|scope sights]] for [[firearm]]s are calibrated to this definition. In addition, three other related definitions are used for artillery and navigation, often called a 'mil', which are ''approximately'' equal to a milliradian. Under these three other definitions, one turn makes up for exactly 6000, 6300, or 6400 mils, spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the milliradian is approximately 0.05729578 degrees (3.43775 minutes). One "[[NATO]] mil" is defined as {{sfrac|6400}} of a turn. Just like with the milliradian, each of the other definitions approximates the milliradian's useful property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1&nbsp;km away ({{sfrac|2{{pi}}|6400}} = 0.0009817... ≈ {{sfrac|1000}}).
| [[Binary angular measurement|binary degree]] || 256 || 1.40625° || The ''binary degree'', also known as the ''[[binary radian]]'' or ''brad'' or ''binary angular measurement'' (''BAM'').<ref name="ooPIC"/>
|-
|-
|[[Binary angular measurement|binary degree]] ||256||1°33′45″  || The ''binary degree'', also known as the ''[[binary radian]]'' or ''brad'' or ''binary angular measurement (BAM)''.<ref name="ooPIC"/> The binary degree is used in computing so that an angle can be efficiently represented in a single [[byte]] (albeit to limited precision). Other measures of the angle used in computing may be based on dividing one whole turn into 2<sup>''n''</sup> equal parts for other values of ''n''.
| [[circular sector|quadrant]] || 4 || 90° || One ''quadrant''{{dubious|reason=A circular sector refers to an area of a figure, not to an angle.|date=June 2025}} is a {{sfrac|4}}&nbsp;turn and also known as a ''[[right angle]]''. In German, the symbol <sup>∟</sup> has been used to denote a right angle.
<ref name="Hargreaves_2010"/> It is {{sfrac|256}} of a turn.<ref name="ooPIC"/>
|-
|-
|{{anchor|Multiples of π}}{{pi}} radian||2||180° || The ''multiples of {{pi}} radians'' (MUL{{pi}}) unit is implemented in the [[Reverse Polish Notation|RPN]] scientific calculator [[WP&nbsp;43S]].<ref name="Bonin_2016"/> See also: [[IEEE 754 recommended operations]]
| [[circular sector|sextant]] || 6 || 60° || The ''sextant'' was the unit used by the [[Babylonians]].<ref name="Jeans_1947"/><ref name="Murnaghan_1946"/>{{dubious|reason=The accessible source does not mention the term, and a circular sector (what the term normally refers to) is an area of a figure, not to an angle.|date=June 2025}}
|-
|-
|[[circular sector|quadrant]]||4||90°||One ''quadrant'' is a {{sfrac|4}}&nbsp;turn and also known as a ''[[right angle]]''. The quadrant is the unit in [[Euclid's Elements]]. In German, the symbol <sup>∟</sup> has been used to denote a quadrant. 1 quad = 90° = {{sfrac|{{pi}}|2}}&nbsp;rad = {{sfrac|4}} turn = 100&nbsp;grad.
| hexacontade || 60 || 6° || The ''hexacontade'' is a unit used by [[Eratosthenes]], with 60 hexacontades in a turn.{{citation needed|date=June 2025}}
|-
|-
|[[circular sector|sextant]]||6||60°||The ''sextant'' was the unit used by the [[Babylonians]],<ref name="Jeans_1947"/><ref name="Murnaghan_1946"/> The degree, minute of arc and second of arc are [[sexagesimal]] subunits of the Babylonian unit. It is straightforward to construct with ruler and compasses. It is the ''angle of the [[equilateral triangle]]'' or is {{sfrac|6}}&nbsp;turn. 1 Babylonian unit = 60° = {{pi}}/3&nbsp;rad ≈ 1.047197551&nbsp;rad.
| diameter part || ~376.991 || ~0.95493° || The ''diameter part'' (occasionally used in Islamic mathematics) is {{sfrac|60}} radian.{{citation needed|date=June 2025}}
|-
|-
| hexacontade||60 ||6°||The ''hexacontade'' is a unit used by [[Eratosthenes]]. It equals 6°, so a whole turn was divided into 60 hexacontades.
| zam || 224 || ~1.607° || In old Arabia, a [[Turn (geometry)|turn]] was subdivided into 32 akhnam, and each akhnam was subdivided into 7 zam.{{citation needed|date=June 2025}}
|-
| [[pechus]]|| 144 to 180 || 2° to 2°30′ || The ''pechus'' was a [[Babylonian mathematics|Babylonian]] unit equal to about 2° or {{sfrac|2|1|2}}°.
|-
| diameter part || ≈376.991 || ≈0.95493° || The ''diameter part'' (occasionally used in Islamic mathematics) is {{sfrac|60}} radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn.
|-
| zam || 224 || ≈1.607° || In old Arabia, a [[Turn (geometry)|turn]] was subdivided into 32 Akhnam, and each akhnam was subdivided into 7 zam so that a [[Turn (geometry)|turn]] is 224 zam.
|}
|}


===Dimensional analysis===
=== Dimensional analysis ===
{{Further|Radian#Dimensional analysis}}
{{Further|Radian#Dimensional analysis}}
In mathematics and the [[International System of Quantities]], an angle is defined as a dimensionless quantity, and in particular, the [[radian]] unit is dimensionless. This convention impacts how angles are treated in [[dimensional analysis]]. For example, when one measures an angle in radians by dividing the arc length by the radius, one is essentially dividing a length by another length, and the units of length cancel each other out. Therefore the result—the angle—doesn't have a physical "dimension" like meters or seconds. This holds true with all angle units, such as radians, degrees, or turns—they all represent a pure number quantifying how much something has turned. This is why, in many equations, angle units seem to "disappear" during calculations, which can sometimes be a bit confusing.
In mathematics and the [[International System of Quantities]], an angle is defined as a dimensionless quantity, and in particular, the [[radian]] is defined as dimensionless in the [[International System of Units]]. This convention prevents angles providing information for [[dimensional analysis]].{{Citation needed|reason=Reliable source(s) needed for the whole paragraph|date=June 2025}}


This disappearing act, while mathematically convenient, has led to significant discussion among scientists and teachers, as it can be tricky to explain and feels inconsistent. To address this, some scientists have suggested treating the angle as having its own fundamental dimension, similar to length or time. This would mean that angle units like radians would always be explicitly present in calculations, making the dimensional analysis more straightforward. However, this approach would also require changing many well-known mathematical and physics formulas, making them longer and perhaps a bit less familiar. For now, the established practice is to consider angles dimensionless, understanding that while units like radians are important for expressing the angle's magnitude, they don't carry a physical dimension in the same way that meters or kilograms do.
While mathematically convenient, this has led to significant discussion among scientists and teachers. Some scientists have suggested treating the angle as having its own dimension, similar to length or time. This would mean that angle units like radians would always be explicitly present in calculations, making the dimensional analysis more straightforward. However, this approach would also require changing many well-known mathematical and physics formulas.{{Citation needed|reason=Reliable source(s) needed for the whole paragraph|date=June 2025}}


===Signed angles ===
=== Signed angles ===
{{main|Angle of rotation}}
{{main|Angle of rotation}}
{{see also|Sign (mathematics)#Angles|Euclidean space#Angle}}
{{see also|Sign (mathematics)#Angles|Euclidean space#Angle}}
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In [[navigation]], [[bearing (navigation)|bearings]] or [[azimuth]] are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.
In [[navigation]], [[bearing (navigation)|bearings]] or [[azimuth]] are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.


===Equivalent angles===
=== Equivalent angles ===
* Angles that have the same measure (i.e., the same magnitude) are said to be ''equal'' or ''[[Congruence (geometry)|congruent]]''. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all ''right angles'' are equal in measure).
* Angles that have the same measure (i.e., the same magnitude) are said to be ''equal'' or ''[[Congruence (geometry)|congruent]]''. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all ''right angles'' are equal in measure).
* Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called ''coterminal angles''.
* Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called ''coterminal angles''.
* The ''reference angle'' (sometimes called ''related angle'') for any angle ''θ'' in standard position is the positive acute angle between the terminal side of ''θ'' and the x-axis (positive or negative).<ref>{{cite web|url=http://www.mathwords.com/r/reference_angle.htm|title=Mathwords: Reference Angle|website=www.mathwords.com|access-date=26 April 2018|url-status=live|archive-url=https://web.archive.org/web/20171023035017/http://www.mathwords.com/r/reference_angle.htm|archive-date=23 October 2017}}</ref><ref>{{cite book |last1=McKeague |first1=Charles P. |title=Trigonometry |date=2008 |publisher=Thomson Brooks/Cole |location=Belmont, CA |isbn=978-0495382607 |page=110 |edition=6th}}</ref> Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude [[modulo]] {{sfrac|2}} turn, 180°, or {{math|π}} radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510° correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).
* The ''reference angle'' (sometimes called ''related angle'') for any angle ''θ'' in standard position is the positive acute angle between the terminal side of ''θ'' and the x-axis (positive or negative).<ref>{{cite web|url=http://www.mathwords.com/r/reference_angle.htm|title=Mathwords: Reference Angle|website=www.mathwords.com|access-date=26 April 2018|url-status=live|archive-url=https://web.archive.org/web/20171023035017/http://www.mathwords.com/r/reference_angle.htm|archive-date=23 October 2017}}</ref><ref>{{cite book |last1=McKeague |first1=Charles P. |title=Trigonometry |date=2008 |publisher=Thomson Brooks/Cole |location=Belmont, CA |isbn=978-0495382607 |page=110 |edition=6th}}</ref> Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude [[modulo]] {{sfrac|2}} turn, 180°, or {{math|π}} radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510° correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).


===Related quantities===
=== Related quantities ===
For an angular unit, it is definitional that the [[angle addition postulate]] holds. Some quantities related to angles where the angle addition postulate does not hold include:
For an angular unit, it is definitional that the [[angle addition postulate]] holds, however some measurements or quantities related to angles are in use that do not satisfy this postulate:
* The ''[[slope]]'' or ''gradient'' is equal to the [[tangent (trigonometric function)|tangent]] of the angle; a gradient is often expressed as a percentage. For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction.
* The ''[[slope]]'' or ''gradient'' is equal to the [[tangent (trigonometric function)|tangent]] of the angle and is often expressed as a percentage ("rise" over "run"). For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction. An [[Grade (slope)|elevation grade]] is a slope used to indicate the steepness of roads, paths and railway lines.
* The ''[[spread (rational trigonometry)|spread]]'' between two lines is defined in [[rational geometry]] as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.
* The ''[[spread (rational trigonometry)|spread]]'' between two lines is defined in [[rational geometry]] as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.
* Although done rarely, one can report the direct results of [[trigonometric functions]], such as the [[sine]] of the angle.
* Although done rarely, one can report the direct results of [[trigonometric functions]], such as the [[sine]] of the angle.


==Angles between curves==
== Angles between curves ==
[[File:Curve angles.svg|thumb|right|The angle between the two curves at ''P'' is defined as the angle between the tangents <var>A</var> and <var>B</var> at <var>P</var>.]]
[[File:Curve angles.svg|thumb|right|The angle between the two curves at P is defined as the angle between the tangents <var>A</var> and <var>B</var> at <var>P</var>.]]
The angle between a line and a [[curve]] (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the [[tangent]]s at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—''amphicyrtic'' (Gr. {{lang|grc|ἀμφί}}, on both sides, κυρτός, convex) or ''cissoidal'' (Gr. κισσός, ivy), biconvex; ''xystroidal'' or ''sistroidal'' (Gr. ξυστρίς, a tool for scraping), concavo-convex; ''amphicoelic'' (Gr. κοίλη, a hollow) or ''angulus lunularis'', biconcave.<ref>{{harvnb|Chisholm|1911}}; {{harvnb|Heiberg|1908|p=178}}</ref><!-- Again, most of this paragraph is from EB1911, with Heath as its source. -->
The angle between a line and a [[curve]] (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the [[tangent]]s at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—''amphicyrtic'' (Gr. {{lang|grc|ἀμφί}}, on both sides, κυρτός, convex) or ''cissoidal'' (Gr. κισσός, ivy), biconvex; ''xystroidal'' or ''sistroidal'' (Gr. ξυστρίς, a tool for scraping), concavo-convex; ''amphicoelic'' (Gr. κοίλη, a hollow) or ''angulus lunularis'', biconcave.<ref>{{harvnb|Chisholm|1911}}; {{harvnb|Heiberg|1908|p=178}}</ref><!-- Again, most of this paragraph is from EB1911, with Heath as its source. -->


==Bisecting and trisecting angles==
== Bisecting and trisecting angles ==
{{Main article|Bisection#Angle bisector|Angle trisection}}
{{Main article|Bisection#Angle bisector|Angle trisection}}


The [[Greek mathematics|ancient Greek mathematicians]] knew how to bisect an angle (divide it into two angles of equal measure) using only a [[compass and straightedge]] but could only trisect certain angles. In 1837, [[Pierre Wantzel]] showed that this construction could not be performed for most angles.
The [[Greek mathematics|ancient Greek mathematicians]] knew how to bisect an angle (divide it into two angles of equal measure) using only a [[compass and straightedge]] but could only trisect certain angles. In 1837, [[Pierre Wantzel]] showed that this construction could not be performed for most angles.


==Dot product and generalisations{{anchor|Dot product}}==
== Dot product and generalisations <span class="anchor" id="Dot product"></span> ==
In the [[Euclidean space]], the angle ''θ'' between two [[Euclidean vector]]s '''u''' and '''v''' is related to their [[dot product]] and their lengths by the formula
In the [[Euclidean space]], the angle ''θ'' between two [[Euclidean vector]]s '''u''' and '''v''' is related to their [[dot product]] and their lengths by the formula
<math display="block"> \mathbf{u} \cdot \mathbf{v} = \cos(\theta) \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>
<math display="block"> \mathbf{u} \cdot \mathbf{v} = \cos(\theta) \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>


This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their [[normal vector]]s and between [[skew lines]] from their vector equations.
This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their [[normal vector]]s and between [[skew lines]] from their vector equations.


===Inner product===
=== Inner product ===
To define angles in an abstract real [[inner product space]], we replace the Euclidean dot product ( '''·''' ) by the inner product <math> \langle \cdot , \cdot \rangle </math>, i.e.
To define angles in an abstract real [[inner product space]], we replace the Euclidean dot product ( '''·''' ) by the inner product <math> \langle \cdot , \cdot \rangle </math>, i.e.
<math display="block"> \langle \mathbf{u} , \mathbf{v} \rangle = \cos(\theta)\ \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>
<math display="block"> \langle \mathbf{u} , \mathbf{v} \rangle = \cos(\theta)\ \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>


In a complex [[inner product space]], the expression for the cosine above may give non-real values, so it is replaced with
In a complex [[inner product space]], the expression for the cosine above may give non-real values, so it is replaced with
<math display="block"> \operatorname{Re} \left( \langle \mathbf{u} , \mathbf{v} \rangle \right) = \cos(\theta) \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>
<math display="block"> \operatorname{Re} \left( \langle \mathbf{u} , \mathbf{v} \rangle \right) = \cos(\theta) \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>
or, more commonly, using the absolute value, with
or, more commonly, using the absolute value, with
<math display="block"> \left| \langle \mathbf{u} , \mathbf{v} \rangle \right| = \left| \cos(\theta) \right| \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>
<math display="block"> \left| \langle \mathbf{u} , \mathbf{v} \rangle \right| = \left| \cos(\theta) \right| \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| .</math>


The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces <math>\operatorname{span}(\mathbf{u})</math> and <math>\operatorname{span}(\mathbf{v})</math> spanned by the vectors <math>\mathbf{u}</math> and <math>\mathbf{v}</math> correspondingly.
The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces <math>\operatorname{span}(\mathbf{u})</math> and <math>\operatorname{span}(\mathbf{v})</math> spanned by the vectors <math>\mathbf{u}</math> and <math>\mathbf{v}</math> correspondingly.


===Angles between subspaces===
=== Angles between subspaces ===
The definition of the angle between one-dimensional subspaces <math>\operatorname{span}(\mathbf{u})</math> and <math>\operatorname{span}(\mathbf{v})</math> given by
The definition of the angle between one-dimensional subspaces <math>\operatorname{span}(\mathbf{u})</math> and <math>\operatorname{span}(\mathbf{v})</math> given by
<math display="block"> \left| \langle \mathbf{u} , \mathbf{v} \rangle \right| = \left| \cos(\theta) \right| \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| </math>
<math display="block"> \left| \langle \mathbf{u} , \mathbf{v} \rangle \right| = \left| \cos(\theta) \right| \left\| \mathbf{u} \right\| \left\| \mathbf{v} \right\| </math>
in a [[Hilbert space]] can be extended to subspaces of finite number of dimensions. Given two subspaces <math> \mathcal{U} </math>, <math> \mathcal{W} </math> with <math> \dim ( \mathcal{U}) := k \leq \dim ( \mathcal{W}) := l </math>, this leads to a definition of <math>k</math> angles called canonical or [[principal angles]] between subspaces.


in a [[Hilbert space]] can be extended to subspaces of finite dimensions. Given two subspaces <math> \mathcal{U} </math>, <math> \mathcal{W} </math> with <math> \dim ( \mathcal{U}) := k \leq \dim ( \mathcal{W}) := l </math>, this leads to a definition of <math>k</math> angles called canonical or [[principal angles]] between subspaces.
=== Angles in Riemannian geometry ===
 
===Angles in Riemannian geometry===
In [[Riemannian geometry]], the [[metric tensor]] is used to define the angle between two [[tangent]]s. Where ''U'' and ''V'' are tangent vectors and ''g''<sub>''ij''</sub> are the components of the metric tensor ''G'',
In [[Riemannian geometry]], the [[metric tensor]] is used to define the angle between two [[tangent]]s. Where ''U'' and ''V'' are tangent vectors and ''g''<sub>''ij''</sub> are the components of the metric tensor ''G'',
<math display="block">
<math display="block">
\cos \theta = \frac{g_{ij} U^i V^j}{\sqrt{ \left| g_{ij} U^i U^j \right| \left| g_{ij} V^i V^j \right|}}.
\cos \theta = \frac{g_{ij} U^i V^j}{\sqrt{ \left| g_{ij} U^i U^j \right| \left| g_{ij} V^i V^j \right|}}.
</math>
</math>


===Hyperbolic angle===
=== Hyperbolic angle ===
A [[hyperbolic angle]] is an [[argument of a function|argument]] of a [[hyperbolic function]] just as the ''circular angle'' is the argument of a [[circular function]]. The comparison can be visualized as the size of the openings of a [[hyperbolic sector]] and a [[circular sector]] since the [[area]]s of these sectors correspond to the angle magnitudes in each case.<ref>[[Robert Baldwin Hayward]] (1892) [https://archive.org/details/algebraofcoplana00haywiala/page/n5/mode/2up ''The Algebra of Coplanar Vectors and Trigonometry''], chapter six</ref> Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as [[infinite series]] in their angle argument, the circular ones are just [[alternating series]] forms of the hyperbolic functions. This comparison of the two series corresponding to functions of angles was described by [[Leonhard Euler]] in ''[[Introduction to the Analysis of the Infinite]]'' (1748).
A [[hyperbolic angle]] is an [[argument of a function|argument]] of a [[hyperbolic function]] just as the ''circular angle'' is the argument of a [[circular function]]. The comparison can be visualized as the size of the openings of a [[hyperbolic sector]] and a [[circular sector]] since the [[area]]s of these sectors correspond to the angle magnitudes in each case.<ref>[[Robert Baldwin Hayward]] (1892) [https://archive.org/details/algebraofcoplana00haywiala/page/n5/mode/2up ''The Algebra of Coplanar Vectors and Trigonometry''], chapter six</ref> Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as [[infinite series]] in their angle argument, the circular ones are just [[alternating series]] forms of the hyperbolic functions. This comparison of the two series corresponding to functions of angles was described by [[Leonhard Euler]] in ''[[Introduction to the Analysis of the Infinite]]'' (1748).


==History and etymology==
== History and etymology ==
The word ''angle'' comes from the [[Latin]] word {{Lang|la|angulus}}, meaning "corner". [[Cognate]] words include the [[Greek language|Greek]] {{lang|grc|ἀγκύλος}} ({{Lang|grc-la|ankylοs}}) meaning "crooked, curved" and the [[English language|English]] word "[[ankle]]". Both are connected with the [[Proto-Indo-European language|Proto-Indo-European]] root ''*ank-'', meaning "to bend" or "bow".<ref>{{harvnb|Slocum|2007}}</ref>
The word ''angle'' comes from the [[Latin]] word {{Lang|la|angulus}}, meaning "corner". [[Cognate]] words include the [[Greek language|Greek]] {{lang|grc|ἀγκύλος}} ({{Lang|grc-la|ankylοs}}) meaning "crooked, curved" and the [[English language|English]] word "[[ankle]]". Both are connected with the [[Proto-Indo-European language|Proto-Indo-European]] root ''*ank-'', meaning "to bend" or "bow".<ref>{{harvnb|Slocum|2007}}</ref>
<!--Note: ἀγκύλος rather than ἀνκύλος is correct; the γκ is a digraph pronounced [ŋk].-->
<!--Note: ἀγκύλος rather than ἀνκύλος is correct; the γκ is a digraph pronounced [ŋk].-->


[[Euclid]] defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician [[Proclus]], an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by [[Eudemus of Rhodes]], who regarded an angle as a deviation from a [[straight line]]; the second, angle as quantity, by [[Carpus of Antioch]], who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship.<ref>{{harvnb|Chisholm|1911}}; {{harvnb|Heiberg|1908|pp=177–178}}</ref><!-- This paragraph is quoted from EB1911, but its source seems to be Heath.-->
[[Euclid]] defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician [[Proclus]], an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by [[Eudemus of Rhodes]], who regarded an angle as a deviation from a [[straight line]]; the second, angle as quantity, by [[Carpus of Antioch]], who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship.<ref>{{harvnb|Chisholm|1911}}; {{harvnb|Heiberg|1908|pp=177–178}}</ref>
 
=== Vertical angle theorem ===
The equality of vertically opposite angles is called the ''vertical angle theorem''. [[Eudemus of Rhodes]] attributed the proof to [[Thales|Thales of Miletus]].<ref>{{cite book |author=Euclid |author-link=Euclid |title=The Elements |title-link=Euclid's Elements}} Proposition I:13.</ref>{{sfn|Shute|Shirk|Porter|1960|pp=25–27}} The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note,{{sfn|Shute|Shirk|Porter|1960|pp=25–27}} when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as:
* All straight angles are equal.
* Equals added to equals are equal.
* Equals subtracted from equals are equal.
 
When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle ''A'' equals ''x'', the measure of angle ''C'' would be {{nowrap|180° − ''x''}}. Similarly, the measure of angle ''D'' would be {{nowrap|180° − ''x''}}. Both angle ''C'' and angle ''D'' have measures equal to {{nowrap|180° − ''x''}} and are congruent. Since angle ''B'' is supplementary to both angles ''C'' and ''D'', either of these angle measures may be used to determine the measure of Angle ''B''. Using the measure of either angle ''C'' or angle ''D'', we find the measure of angle ''B'' to be {{nowrap|1=180° − (180° − ''x'') = 180° − 180° + ''x'' = ''x''}}. Therefore, both angle ''A'' and angle ''B'' have measures equal to ''x'' and are equal in measure.


==Angles in geography and astronomy==
== Angles in geography and astronomy ==
In [[geography]], the location of any point on the Earth can be identified using a ''[[geographic coordinate system]]''. This system specifies the [[latitude]] and [[longitude]] of any location in terms of angles subtended at the center of the Earth, using the [[equator]] and (usually) the [[Greenwich meridian]] as references.
In [[geography]], the location of any point on the Earth can be identified using a ''[[geographic coordinate system]]''. This system specifies the [[latitude]] and [[longitude]] of any location in terms of angles subtended at the center of the Earth, using the [[equator]] and (usually) the [[Greenwich meridian]] as references.


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|-
|-
! Minute
! Minute
| m || 0°15′ || {{frac|{{pi}}|720}} rad ||{{frac|1|1,440}} turn || {{frac|1|60}} hour  
| m || 0°15′ || {{frac|{{pi}}|720}} rad ||{{frac|1|{{val|1440}}}} turn || {{frac|1|60}} hour  
|-
|-
! Second
! Second
| s || 0°0′15″ || {{frac|{{pi}}|43200}} rad || {{frac|1|86,400}} turn || {{frac|1|60}} minute
| s || 0°0′15″ || {{frac|{{pi}}|43200}} rad || {{frac|1|{{val|86400}}}} turn || {{frac|1|60}} minute
|}
|}


==See also==
== See also ==
{{div col |colwidth=22em}}
{{div col |colwidth=22em}}
* [[Angle measuring instrument]]
* [[Angle measuring instrument]]
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{{div col end}}
{{div col end}}


==Notes==
== Notes ==
{{Reflist|group="nb"}}
{{reflist|group="nb"}}
{{notelist}}
{{notelist}}


==References==
== References ==
{{Reflist|refs=
{{reflist|refs=
<ref name="Jeans_1947">{{cite book |author-first=James Hopwood |author-last=Jeans |author-link=James Hopwood Jeans |date=1947 |title=The Growth of Physical Science |publisher=CUP Archive |page=[https://archive.org/details/in.ernet.dli.2015.210060/page/n25 7] |url=https://archive.org/details/in.ernet.dli.2015.210060}}</ref>
<ref name="Jeans_1947">{{cite book |author-first=James Hopwood |author-last=Jeans |author-link=James Hopwood Jeans |date=1947 |title=The Growth of Physical Science |publisher=CUP Archive |page=[https://archive.org/details/in.ernet.dli.2015.210060/page/n25 7] |url=https://archive.org/details/in.ernet.dli.2015.210060}}</ref>
<ref name="Murnaghan_1946">{{cite book |author-first=Francis Dominic |author-last=Murnaghan |author-link=Francis Dominic Murnaghan (mathematician) |date=1946 |title=Analytic Geometry |page=2}}</ref>
<ref name="Murnaghan_1946">{{cite book |author-first=Francis Dominic |author-last=Murnaghan |author-link=Francis Dominic Murnaghan (mathematician) |date=1946 |title=Analytic Geometry |page=2}}</ref>
<ref name="Dimitric_2012">{{cite journal |title=On Angles and Angle Measurements |author-first=Radoslav M. |author-last=Dimitrić |journal=The Teaching of Mathematics |date=2012 |volume=XV |number=2 |pages=133–140 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/29/tm1525.pdf |access-date=2019-08-06 |url-status=live |archive-url=https://web.archive.org/web/20190117195213/http://elib.mi.sanu.ac.rs/files/journals/tm/29/tm1525.pdf |archive-date=2019-01-17}}</ref>
<ref name="Dimitric_2012">{{cite journal |title=On Angles and Angle Measurements |author-first=Radoslav M. |author-last=Dimitrić |journal=The Teaching of Mathematics |date=2012 |volume=XV |number=2 |pages=133–140 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/29/tm1525.pdf |access-date=2019-08-06 |url-status=live |archive-url=https://web.archive.org/web/20190117195213/http://elib.mi.sanu.ac.rs/files/journals/tm/29/tm1525.pdf |archive-date=2019-01-17}}</ref>
<ref name="ooPIC">{{cite web |title=ooPIC Programmer's Guide - Chapter 15: URCP |work=[[ooPIC]] Manual & Technical Specifications - ooPIC Compiler Ver 6.0 |orig-year=1997 |date=2007 |publisher=Savage Innovations, LLC |url=http://www.oopic.com/pgchap15.htm |access-date=2019-08-05 |url-status=dead |archive-url=https://web.archive.org/web/20080628051746/http://www.oopic.com/pgchap15.htm |archive-date=2008-06-28}}</ref>
<ref name="ooPIC">{{cite web |title=ooPIC Programmer's Guide - Chapter 15: URCP |work=[[ooPIC]] Manual & Technical Specifications - ooPIC Compiler Ver 6.0 |orig-year=1997 |date=2007 |publisher=Savage Innovations, LLC |url=http://www.oopic.com/pgchap15.htm |access-date=2019-08-05 |url-status=dead |archive-url=https://web.archive.org/web/20080628051746/http://www.oopic.com/pgchap15.htm |archive-date=2008-06-28}}</ref>
<ref name="Hargreaves_2010">{{cite web |title=Angles, integers, and modulo arithmetic |author-first=Shawn |author-last=Hargreaves |author-link=:pl:Shawn Hargreaves |publisher=blogs.msdn.com |url=http://blogs.msdn.com/shawnhar/archive/2010/01/04/angles-integers-and-modulo-arithmetic.aspx |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20190630223817/http://www.shawnhargreaves.com/blogindex.html |archive-date=2019-06-30}}</ref>
<ref name="Bonin_2016">{{cite web |title=RE: WP-32S in 2016? |date=2016-01-11 |author-first=Walter |author-last=Bonin |work=HP Museum |url=https://www.hpmuseum.org/forum/thread-5427-post-48957.html#pid48957 |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20190806141349/https://www.hpmuseum.org/forum/thread-5427-post-48957.html |archive-date=2019-08-06}}</ref>
}}
}}


== Bibliography ==
== Bibliography ==
{{refbegin|30em}}
* {{citation
* {{citation
  | last = Aboughantous | first = Charles H.  
  | last = Aboughantous | first = Charles H.  
Line 459: Line 397:
* {{cite journal |last1=Romain |first1=Jacques E. |title=Angle as a fourth fundamental quantity |journal=Journal of Research of the National Bureau of Standards Section B |date=July 1962 |volume=66B |issue=3 |pages=97 |doi=10.6028/jres.066B.012 |doi-access=free}}
* {{cite journal |last1=Romain |first1=Jacques E. |title=Angle as a fourth fundamental quantity |journal=Journal of Research of the National Bureau of Standards Section B |date=July 1962 |volume=66B |issue=3 |pages=97 |doi=10.6028/jres.066B.012 |doi-access=free}}
* {{SpringerEOM|mode=cs2 |last=Sidorov |first=L. A. |date=2001 |id=Angle&oldid=13323 |title=Angle}}
* {{SpringerEOM|mode=cs2 |last=Sidorov |first=L. A. |date=2001 |id=Angle&oldid=13323 |title=Angle}}
* {{citation|last=Slocum|first=Jonathan|date=2007|url=http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html|title=Preliminary Indo-European lexicon&nbsp;— Pokorny PIE data|access-date=2 Feb 2010|publisher=[[Linguistics Research Center at UT Austin|University of Texas research department: linguistics research center]]|archive-date=27 June 2010|archive-url=https://web.archive.org/web/20100627012240/http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html|url-status=dead}}
* {{citation |last=Slocum |first=Jonathan |date=2007 |url=http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html |title=Preliminary Indo-European lexicon&nbsp;— Pokorny PIE data |access-date=2 Feb 2010 |publisher=[[Linguistics Research Center at UT Austin|University of Texas research department: linguistics research center]] |archive-date=27 June 2010 |archive-url=https://web.archive.org/web/20100627012240/http://www.utexas.edu/cola/centers/lrc/ielex/X/P0089.html |url-status=dead}}
* {{citation |last1=Shute |first1=William G. |last2=Shirk |first2=William W. |last3=Porter |first3=George F. |date=1960 |title=Plane and Solid Geometry |publisher=American Book Company  |pages=25–27}}
* {{citation |last1=Shute |first1=William G. |last2=Shirk |first2=William W. |last3=Porter |first3=George F. |date=1960 |title=Plane and Solid Geometry |publisher=American Book Company  |pages=25–27}}
* {{cite journal |last1=Torrens |first1=A B |title=On Angles and Angular Quantities |journal=Metrologia |date=1 January 1986 |volume=22 |issue=1 |pages=1–7 |doi=10.1088/0026-1394/22/1/002 |bibcode=1986Metro..22....1T |s2cid=250801509}}
* {{cite journal |last1=Torrens |first1=A B |title=On Angles and Angular Quantities |journal=Metrologia |date=1 January 1986 |volume=22 |issue=1 |pages=1–7 |doi=10.1088/0026-1394/22/1/002 |bibcode=1986Metro..22....1T |s2cid=250801509}}
* {{citation |last1=Wong |first1=Tak-wah |last2=Wong |first2=Ming-sim |date=2009 |chapter=Angles in Intersecting and Parallel Lines |title=New Century Mathematics |location=Hong Kong |publisher=Oxford University Press |edition=1 |volume=1B |pages=161–163  |isbn=978-0-19-800177-5}}
* {{citation |last1=Wong |first1=Tak-wah |last2=Wong |first2=Ming-sim |date=2009 |chapter=Angles in Intersecting and Parallel Lines |title=New Century Mathematics |location=Hong Kong |publisher=Oxford University Press |edition=1 |volume=1B |pages=161–163  |isbn=978-0-19-800177-5}}
{{EB1911 |wstitle=Angle |volume=2 |page=14 |mode=cs2}}
{{EB1911 |wstitle=Angle |volume=2 |page=14 |mode=cs2}}
{{refend}}


==External links==
== External links ==
{{Commons category|Angles (geometry)}}
{{Commons category|Angles (geometry)}}
{{Wikibooks|Geometry|Unified Angles}}  
{{Wikibooks|Geometry|Unified Angles}}  

Revision as of 11:25, 15 June 2025

Template:Short description Script error: No such module "Distinguish". Script error: No such module "about".

two line bent at a point
A green angle formed by two red rays on the Cartesian coordinate system

In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.[1][2] More generally angles are also formed wherever two lines, rays or line segments come together, such as at the corners of triangles and other polygons. An angle can be considered as the region of the plane bounded by the sides.[3][4]Template:Efn Angles can also be formed by the intersection of two planes or by two intersecting curves, in which case the rays lying tangent to each curve at the point of intersection define the angle.

The term angle is also used for the size, magnitude or quantity of these types of geometric figures and in this context an angle consists of a number and unit of measurement. Angular measure or measure of angle are sometimes used to distinguish between the measurement and figure itself. The measurement of angles is intrinsically linked with circles and rotation. For an ordinary angle, this is often visualized or defined using the arc of a circle centered at the vertex and lying between the sides.

Fundamentals

An angle is a figure lying in a plane formed by two distinct rays (half-lines emanating indefinitely from an endpoint in one direction), which share a common endpoint. The rays are called the sides or arms of the angle, and the common endpoint is called the vertex. The sides divide the plane into two regions: the interior of the angle and the exterior of the angle.[1]

Notation

File:Angle diagram.svg
∠BAC is formed by rays AB→ and AC→. θ is the conventional measure of ∠BAC and β is an alternative measure.

An angle symbol (∠ or ^, read as "angle") together with one or three defining points is used to identify angles in geometric figures. For example, the angle with vertex A formed by the rays AB→ and AC→ is denoted as ∠A (using the vertex alone) or ∠BAC (with the vertex always named in the middle). The size or measure of the angle is denoted m∠A or m∠BAC.

In geometric figures and mathematical expressions, it is also common to use Greek letters (α, β, γ, θ, φ, ...) or lower case Roman letters (a, b, c, ...) as variables to represent the size of an angle.Template:Sfn

Conventionally, angle size is measured "between" the sides through the interior of the angle and given as a magnitude or scalar quantity. At other times it might be measured through the exterior of the angle or given as a signed number to indicate a direction of measurement.Script error: No such module "Unsubst".

Units of measurement

Angles are measured in various units, the most common being the degree (denoted by the symbol °), radian (denoted by the symbol rad) and turn. These units differ in the way they divide up a full angle, an angle where one ray, initially congruent to the other, performs a compete rotation about the vertex to return back to its starting position.[5]

Degrees and turns are defined directly with reference to a full angle, which measures 1 turn or 360°.[6] A measure in turns gives an angle's size as a proportion of a full angle and a degree can be considered as a subdivision of a turn. Radians are not defined directly in relation to a full angle (see Template:Section link), but in such a way that its measure is 2Template:Math rad, approximately 6.28 rad.[7]

File:Angle types.svg
Common types of angles

Common angles

  • An angle equal to 0° or not turned is called a zero angle.Template:Sfn
  • An angle smaller than a right angle (less than 90°) is called an acute angle.Template:Sfn
  • An angle equal to Template:Sfrac turn (90° or Template:Sfrac rad) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular.Template:Sfn
  • An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an obtuse angleTemplate:Sfn ("obtuse" meaning "blunt").
  • An angle equal to Template:Sfrac turn (180° or Template:Math rad) is called a straight angle.Template:Sfn
  • An angle larger than a straight angle but less than 1 turn (between 180° and 360°) is called a reflex angle.
  • An angle equal to 1 turn (360° or 2Template:Math rad) is called a full angle, complete angle, round angle or perigon.
  • An angle that is not a multiple of a right angle is called an oblique angle.

The names, intervals, and measuring units are shown in the table below:

Name   zero angle acute angle right angle obtuse angle straight angle reflex angle full angle
Unit Interval
turn   0 turn (0, Template:Sfrac) turn Template:Sfrac turn (Template:Sfrac, Template:Sfrac) turn Template:Sfrac turn (Template:Sfrac, 1) turn 1 turn
degree   0° (0, 90)° 90° (90, 180)° 180° (180, 360)° 360°
radian 0 rad (0, Template:SfracTemplate:Pi) rad Template:SfracTemplate:Pi rad (Template:SfracTemplate:Pi, Template:Pi) rad Template:Pi rad (Template:Pi, 2Template:Pi) rad 2Template:Pi rad

Addition and subtraction

File:Angle addition.svg
The angle addition postulate defines addition and subtraction of angles: θ + α = φ; φ − α = θ.

The angle addition postulate states that if D is a point lying in the interior of ∠BAC then:[8] m∠BAC=m∠BAD+m∠DAC. This relationship defines what it means add any two angles: their vertices are placed together while sharing a side to create a new larger angle. The measure of the new larger angle is the sum of the measures of the two angles. Subtraction follows from rearrangement of the formula.Script error: No such module "Unsubst".

Types

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Adjacent and vertical angles

File:Adjacentangles.svg
Angles A and B are adjacent.

Template:Redirect-distinguish

Adjacent angles (abbreviated adj. ∠s), are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively called complementary, supplementary, and explementary angles (see Template:Section link below).

File:Vertical Angles.svg
Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles. Hatch marks are used here to show angle equality.

Vertical angles are formed when two straight lines intersect at a point producing four angles. A pair of angles opposite each other are called vertical angles, opposite angles or vertically opposite angles (abbreviated vert. opp. ∠s),[9] where "vertical" refers to the sharing of a vertex, rather than an up-down orientation. The vertical angle theorem states that vertical angles are always congruent or equal to each other.Script error: No such module "Unsubst".

A transversal is a line that intersects a pair of (often parallel) lines and is associated with exterior angles, interior angles, alternate exterior angles, alternate interior angles, corresponding angles, and consecutive interior angles.Template:Sfn

Combining angle pairs

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When summing two angles that are either adjacent or separated in space, three cases are of particular importance.Script error: No such module "Unsubst".

Complementary angles

File:Complement angle.svg
Angles a and b are complementary angles

Complementary angles are angle pairs whose measures sum to a right angle (Template:Sfrac turn, 90°, or Template:Sfrac rad).[10] If the two complementary angles are adjacent, their non-shared sides form a right angle. In a right-angle triangle the two acute angles are complementary as the sum of the internal angles of a triangle is 180°.

The difference between an angle and a right angle is termed the complement of the angle[11] which is from the Latin complementum and associated verb complere, meaning "to fill up". An acute angle is "filled up" by its complement to form a right angle.Script error: No such module "Unsubst".


Supplementary angles

File:Angle obtuse acute straight.svg
Angles a and b are supplementary angles

Two angles that sum to a straight angle (Template:Sfrac turn, 180°, or Template:Math rad) are called supplementary angles.[12] If the two supplementary angles are adjacent, their non-shared sides form a straight angle or straight line and are called a linear pair of angles.Template:Sfn The difference between an angle and a straight angle is termed the supplement of the angle.[13]

Examples of non-adjacent complementary angles include the consecutive angles of a parallelogram and opposite angles of a cyclic quadrilateral. For a circle with center O, and tangent lines from an exterior point P touching the circle at points T and Q, the resulting angles ∠TPQ and ∠TOQ are supplementary.

Explementary angles

File:Conjugate Angles.svg
Angles AOB and COD are explementary or conjugate angles

Two angles that sum to a full angle (1 turn, 360°, or 2Template:Math radians) are called explementary angles or conjugate angles.[14] The difference between an angle and a full angle is termed the explement or conjugate of the angle.Script error: No such module "Unsubst".

Polygon-related angles

File:ExternalAngles.svg
Internal and external angles
  • An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon. A simple concave polygon has at least one interior angle, that is, a reflex angle. Template:Pb In Euclidean geometry, the measures of the interior angles of a triangle add up to Template:Math radians, 180°, or Template:Sfrac turn; the measures of the interior angles of a simple convex quadrilateral add up to 2Template:Math radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex polygon with n sides add up to (n − 2)Template:Math radians, or (n − 2)180 degrees, (n − 2)2 right angles, or (n − 2)Template:Sfrac turn.
  • The supplement of an interior angle is called an exterior angle; that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical and hence are equal. An exterior angle measures the amount of rotation one must make at a vertex to trace the polygon.Template:Sfn If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure. Template:Pb In Euclidean geometry, the sum of the exterior angles of a simple convex polygon, if only one of the two exterior angles is assumed at each vertex, will be one full turn (360°). The exterior angle here could be called a supplementary exterior angle. Exterior angles are commonly used in Logo Turtle programs when drawing regular polygons.
  • In a triangle, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point).[15]Template:Rp
  • In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.[15]Template:Rp
  • In a triangle, three intersection points, two between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended are collinear.[15]Template:Rp
  • Some authors use the name exterior angle of a simple polygon to mean the explement exterior angle (not supplement!) of the interior angle.[16] This conflicts with the above usage.

Plane-related angles

  • The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle.[11] It may be defined as the acute angle between two lines normal to the planes.
  • The angle between a plane and an intersecting straight line is complementary to the angle between the intersecting line and the normal to the plane.

Measuring angles

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File:Basic angle in circle.svg
The angle size θ can be measured as s/r radians or s/C turns

Measurement of angles is intrinsically linked with circles and rotation. An angle is measured by placing it within a circle of any size, with the vertex at the circle's centre and the sides intersecting the perimeter.

An arc s is formed as the shortest distance on the perimeter between the two points of intersection, which is said to be the arc subtended by the angle.

The length of s can be used to measure the angle's size θ, however as s is dependent on the size of the circle chosen, it must be adjusted so that any arbitrary circle will give the same measure of angle. This can be done in two ways: by taking the ratio to either the radius r or circumference C of the circle.

The ratio of the length s by the radius r is the number of radians in the angle, while the ratio of length s by the circumference C is the number of turns:[17] θrad=srradθturn=sC =s2πrturns

File:Angle measure.svg
The measure of angle Template:Math is Template:Sfrac radians.

The value of Template:Math thus defined is independent of the size of the circle: if the length of the radius is changed, then both the circumference and the arc length change in the same proportion, so the ratios Template:Sfrac and Template:Sfrac are unaltered.Template:Refn

Angles of the same size are said to be equal congruent or equal in measure.

Units

In addition to the radian and turn, other angular units exist, typically based on subdivisions of the turn, including the degree (°) and the gradian (grad), though many others have been used throughout history.[18]

Conversion between units may be obtained by multiplying the angular measure in one unit by a conversion constant of the form kakb where ka and kb are the measures of a complete turn in units a and b. For example, to convert an angle of π2radians to degrees: θdeg=kdegkrad⋅θrad=360∘2πrad⋅π2rad=90∘

The following table lists some units used to represent angles.

Name (symbol) Number in one turn 1 unit in degrees Description
turn 1 360° The turn is the angle subtended by the circumference of a circle at its centre. A turn is equal to 2Template:Pi or [[Tau (mathematics)|Template:Tau]] radians.
degree ( ° ) 360 1° The degree is a sexagesimal subunit of the sextant, making one turn equal to 360°.
radian (rad) Template:Math 57.2957...° The radian is the angle subtended by an arc of a circle that has the same length as the circle's radius.
grad (gon) 400 0.9° The grad, also called grade, gradian, or gon, is defined as Template:Sfrac of a right angle. The grad is used mostly in triangulation and continental surveying.
arcminute ( ′ ) 21,600 Template:Sfrac° The minute of arc (or arcminute, or just minute) is a sexagesimal subunit of a degree.
arcsecond ( ″ ) 1,296,000 Template:Sfrac° The second of arc (or arcsecond, or just second) is a sexagesimal subunit of a minute of arc.
milliradian (mrad) Template:Math ~0.0573° The milliradian is a thousandth of a radian. For artillery and navigation a unit is used, often called a 'mil', which are approximately equal to a milliradian. One turn is exactly 6000, 6300, or 6400 mils, depending on which definition is used.
(compass) point 32 11.25° The point or wind, used in navigation, divides the compass (one turn) into 32 points or compass directions.
binary degree 256 1.40625° The binary degree, also known as the binary radian or brad or binary angular measurement (BAM).[19]
quadrant 4 90° One quadrantScript error: No such module "Unsubst". is a Template:Sfrac turn and also known as a right angle. In German, the symbol ∟ has been used to denote a right angle.
sextant 6 60° The sextant was the unit used by the Babylonians.[20][21]Script error: No such module "Unsubst".
hexacontade 60 6° The hexacontade is a unit used by Eratosthenes, with 60 hexacontades in a turn.Script error: No such module "Unsubst".
diameter part ~376.991 ~0.95493° The diameter part (occasionally used in Islamic mathematics) is Template:Sfrac radian.Script error: No such module "Unsubst".
zam 224 ~1.607° In old Arabia, a turn was subdivided into 32 akhnam, and each akhnam was subdivided into 7 zam.Script error: No such module "Unsubst".

Dimensional analysis

Script error: No such module "labelled list hatnote". In mathematics and the International System of Quantities, an angle is defined as a dimensionless quantity, and in particular, the radian is defined as dimensionless in the International System of Units. This convention prevents angles providing information for dimensional analysis.Script error: No such module "Unsubst".

While mathematically convenient, this has led to significant discussion among scientists and teachers. Some scientists have suggested treating the angle as having its own dimension, similar to length or time. This would mean that angle units like radians would always be explicitly present in calculations, making the dimensional analysis more straightforward. However, this approach would also require changing many well-known mathematical and physics formulas.Script error: No such module "Unsubst".

Signed angles

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File:Angles on the unit circle.svg
Measuring from the x-axis, angles on the unit circle count as positive in the counterclockwise direction, and negative in the clockwise direction.

An angle denoted as Template:Math might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, It is therefore frequently helpful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions or "sense" relative to some reference.

In a two-dimensional Cartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive x-axis, while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns, with positive angles representing rotations toward the positive y-axis and negative angles representing rotations toward the negative y-axis. When Cartesian coordinates are represented by standard position, defined by the x-axis rightward and the y-axis upward, positive rotations are anticlockwise, and negative cycles are clockwise.

In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minus θ". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor).

In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an orientation, which is typically determined by a normal vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.

In navigation, bearings or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.

Equivalent angles

  • Angles that have the same measure (i.e., the same magnitude) are said to be equal or congruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all right angles are equal in measure).
  • Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles.
  • The reference angle (sometimes called related angle) for any angle θ in standard position is the positive acute angle between the terminal side of θ and the x-axis (positive or negative).[22][23] Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude modulo Template:Sfrac turn, 180°, or Template:Math radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510° correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).

Related quantities

For an angular unit, it is definitional that the angle addition postulate holds, however some measurements or quantities related to angles are in use that do not satisfy this postulate:

  • The slope or gradient is equal to the tangent of the angle and is often expressed as a percentage ("rise" over "run"). For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction. An elevation grade is a slope used to indicate the steepness of roads, paths and railway lines.
  • The spread between two lines is defined in rational geometry as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.
  • Although done rarely, one can report the direct results of trigonometric functions, such as the sine of the angle.

Angles between curves

File:Curve angles.svg
The angle between the two curves at P is defined as the angle between the tangents A and B at P.

The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. Script error: No such module "Lang"., on both sides, κυρτός, convex) or cissoidal (Gr. κισσός, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίς, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.[24]

Bisecting and trisecting angles

Template:Main article

The ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only a compass and straightedge but could only trisect certain angles. In 1837, Pierre Wantzel showed that this construction could not be performed for most angles.

Dot product and generalisations

In the Euclidean space, the angle θ between two Euclidean vectors u and v is related to their dot product and their lengths by the formula 𝐮⋅𝐯=cos⁡(θ)‖𝐮‖‖𝐯‖.

This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their normal vectors and between skew lines from their vector equations.

Inner product

To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product ⟨⋅,⋅⟩, i.e. ⟨𝐮,𝐯⟩=cos⁡(θ) ‖𝐮‖‖𝐯‖.

In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with Re⁡(⟨𝐮,𝐯⟩)=cos⁡(θ)‖𝐮‖‖𝐯‖. or, more commonly, using the absolute value, with |⟨𝐮,𝐯⟩|=|cos⁡(θ)|‖𝐮‖‖𝐯‖.

The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces span⁡(𝐮) and span⁡(𝐯) spanned by the vectors 𝐮 and 𝐯 correspondingly.

Angles between subspaces

The definition of the angle between one-dimensional subspaces span⁡(𝐮) and span⁡(𝐯) given by |⟨𝐮,𝐯⟩|=|cos⁡(θ)|‖𝐮‖‖𝐯‖ in a Hilbert space can be extended to subspaces of finite number of dimensions. Given two subspaces 𝒰, 𝒲 with dim⁡(𝒰):=k≤dim⁡(𝒲):=l, this leads to a definition of k angles called canonical or principal angles between subspaces.

Angles in Riemannian geometry

In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and gij are the components of the metric tensor G, cos⁡θ=gijUiVj|gijUiUj||gijViVj|.

Hyperbolic angle

A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument of a circular function. The comparison can be visualized as the size of the openings of a hyperbolic sector and a circular sector since the areas of these sectors correspond to the angle magnitudes in each case.[25] Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as infinite series in their angle argument, the circular ones are just alternating series forms of the hyperbolic functions. This comparison of the two series corresponding to functions of angles was described by Leonhard Euler in Introduction to the Analysis of the Infinite (1748).

History and etymology

The word angle comes from the Latin word Script error: No such module "Lang"., meaning "corner". Cognate words include the Greek Script error: No such module "Lang". (Script error: No such module "Lang".) meaning "crooked, curved" and the English word "ankle". Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".[26]

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician Proclus, an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by Eudemus of Rhodes, who regarded an angle as a deviation from a straight line; the second, angle as quantity, by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship.[27]

Vertical angle theorem

The equality of vertically opposite angles is called the vertical angle theorem. Eudemus of Rhodes attributed the proof to Thales of Miletus.[28]Template:Sfn The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note,Template:Sfn when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as:

  • All straight angles are equal.
  • Equals added to equals are equal.
  • Equals subtracted from equals are equal.

When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle A equals x, the measure of angle C would be 180° − x. Similarly, the measure of angle D would be 180° − x. Both angle C and angle D have measures equal to 180° − x and are congruent. Since angle B is supplementary to both angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either angle C or angle D, we find the measure of angle B to be 180° − (180° − x) = 180° − 180° + x = x. Therefore, both angle A and angle B have measures equal to x and are equal in measure.

Angles in geography and astronomy

In geography, the location of any point on the Earth can be identified using a geographic coordinate system. This system specifies the latitude and longitude of any location in terms of angles subtended at the center of the Earth, using the equator and (usually) the Greenwich meridian as references.

In astronomy, a given point on the celestial sphere (that is, the apparent position of an astronomical object) can be identified using any of several astronomical coordinate systems, where the references vary according to the particular system. Astronomers measure the angular separation of two stars by imagining two lines through the center of the Earth, each intersecting one of the stars. The angle between those lines and the angular separation between the two stars can be measured.

In both geography and astronomy, a sighting direction can be specified in terms of a vertical angle such as altitude /elevation with respect to the horizon as well as the azimuth with respect to north.

Astronomers also measure objects' apparent size as an angular diameter. For example, the full moon has an angular diameter of approximately 0.5° when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The small-angle formula can convert such an angular measurement into a distance/size ratio.

Other astronomical approximations include:

  • 0.5° is the approximate diameter of the Sun and of the Moon as viewed from Earth.
  • 1° is the approximate width of the little finger at arm's length.
  • 10° is the approximate width of a closed fist at arm's length.
  • 20° is the approximate width of a handspan at arm's length.

These measurements depend on the individual subject, and the above should be treated as rough rule of thumb approximations only.

In astronomy, right ascension and declination are usually measured in angular units, expressed in terms of time, based on a 24-hour day.

Unit Symbol Degrees Radians Turns Other
Hour h 15° <templatestyles src="Fraction/styles.css" />Template:Pi⁄12 rad <templatestyles src="Fraction/styles.css" />1⁄24 turn
Minute m 0°15′ <templatestyles src="Fraction/styles.css" />Template:Pi⁄720 rad <templatestyles src="Fraction/styles.css" />1⁄Template:Val turn <templatestyles src="Fraction/styles.css" />1⁄60 hour
Second s 0°0′15″ <templatestyles src="Fraction/styles.css" />Template:Pi⁄43200 rad <templatestyles src="Fraction/styles.css" />1⁄Template:Val turn <templatestyles src="Fraction/styles.css" />1⁄60 minute

See also

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Notes

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References

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Bibliography

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  1. ↑ a b Script error: No such module "citation/CS1".
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  15. ↑ a b c Johnson, Roger A. Advanced Euclidean Geometry, Dover Publications, 2007.
  16. ↑ Script error: No such module "citation/CS1". as cited in Script error: No such module "Template wrapper".
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  21. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Murnaghan_1946
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  25. ↑ Robert Baldwin Hayward (1892) The Algebra of Coplanar Vectors and Trigonometry, chapter six
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  28. ↑ Script error: No such module "citation/CS1". Proposition I:13.
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