View source for Trapezoid

{{Short description|Convex quadrilateral with at least one pair of parallel sides}}
{{Use American English|date=August 2020}}
{{Infobox polygon
| name        =  Trapezoid {{small|(American English)}}<br>Trapezium {{small|(British English)}}
| image       = Trapezoid.svg
| caption     =  Trapezoid or trapezium
| type        = [[quadrilateral]]
| edges       = 4
| symmetry    = 
| area        = <math>\tfrac12(a + b) h</math>
| properties  = [[convex polygon|convex]]
}}
{{wikt | trapezoid}}
<!-- Please do not change the definition; this article uses the inclusive definition (at least one pair of parallel sides) and NOT the exclusive definition (only one pair).-->
In [[geometry]], a '''trapezoid''' ({{IPAc-en|ˈ|t|ɹ|æ|p|ə|z|ɔɪ|d}}) in [[North American English]], or '''trapezium''' ({{IPAc-en|t|r|ə|ˈ|p|iː|z|i|ə|m}}) in [[British English]],<ref>{{Cite web |title=Trapezoid – math word definition – Math Open Reference |url=https://www.mathopenref.com/trapezoid.html |access-date=2024-05-15 |website=www.mathopenref.com}}</ref><ref>{{cite book|author1-first=Anthony D.|author1-last=Gardiner|author2-first=Christopher J.|author2-last=Bradley|title=Plane Euclidean Geometry: Theory and Problems|publisher=United Kingdom Mathematics Trust|year=2005|isbn=9780953682362 |page=34}}</ref> is a [[quadrilateral]] that has at least one pair of [[Parallel (geometry)|parallel]] sides.

The parallel sides are called the ''bases'' of the trapezoid.{{sfn|Hopkins|1891|p=33}} The other two sides are called the ''legs''{{sfn|Hopkins|1891|p=33}} or ''lateral sides''. (If the trapezoid is a [[parallelogram]], then the choice of bases and legs is arbitrary.)

A trapezoid is usually considered to be a [[Convex polygon|convex]] quadrilateral in [[Euclidean geometry]], but there are also [[Crossed polygon|crossed]] cases. If ''ABCD'' is a convex trapezoid, then ''ABDC'' is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids.

== Definitions ==
''Trapezoid'' can be defined exclusively or inclusively. Under an exclusive definition a trapezoid is a quadrilateral having {{em|exactly one}} pair of parallel sides, with the other pair of opposite sides non-parallel. Parallelograms including rhombi, rectangles, and squares are then not considered to be trapezoids.{{sfn|Usiskin|Griffin|2008|p=[https://books.google.com/books?id=ff0nDwAAQBAJ&pg=PA29 29]}}{{sfn|Alsina|Nelsen|2020|p=[http://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA90 90]}} Under an inclusive definition, a trapezoid is any quadrilateral with {{em|at least}} one pair of parallel sides.<ref name=ringenberg>{{cite book|title=College Geometry|author1-first=Lawrence A.|author1-last=Ringenberg|publisher=R. E. Krieger Publishing Company|year=1977|isbn=9780882755458|chapter=Coordinates in a Plane |pages=161–162}}</ref> In an inclusive classification scheme, definitions are hierarchical: a square is a type of rectangle and a type of rhombus, a rectangle or rhombus is a type of parallelogram, and every parallelogram is a type of trapezoid.{{sfn|Alsina|Nelsen|2020|p=[http://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA89 89]}}

Professional mathematicians and [[post-secondary education|post-secondary]] geometry textbooks nearly always prefer inclusive definitions and classifications, because they simplify statements and proofs of geometric theorems.{{sfn|Usiskin|Griffin|2008|p=[https://books.google.com/books?id=ff0nDwAAQBAJ&pg=PA32 32]}} In primary and secondary education, definitions of ''rectangle'' and ''parallelogram'' are also nearly always inclusive, but an exclusive definition of ''trapezoid'' is commonly found.<ref>{{cite journal |title=A Quadrilateral Hierarchy to Facilitate Learning in Geometry |first1=Timothy V. |last1=Craine |first2=Rheta N. |last2=Rubenstein |journal=The Mathematics Teacher |year=1993 |volume=86 |number=1 |pages=30–36 |jstor=27968085 |doi=10.5951/MT.86.1.0030 }}</ref><ref>{{cite journal |last=Popovic |first=Gorjana |title=Who is This Trapezoid, Anyway? |year=2012 |doi=10.5951/mathteacmiddscho.18.4.0196 |jstor=10.5951/mathteacmiddscho.18.4.0196 |journal=Mathematics Teaching in the Middle School |volume=18 |number=4 |pages=196–199 |id={{ResearchGatePub|259750174}} }}</ref> This article uses the inclusive definition and considers parallelograms to be special kinds of trapezoids. (Cf. {{slink|Quadrilateral#Taxonomy}}.)

To avoid confusion, some sources use the term ''proper trapezoid'' to describe trapezoids with exactly one pair of parallel sides, analogous to uses of the word ''proper'' in some other mathematical objects.<ref>{{cite web|last=Michon|first=Gérard P.|url=http://www.numericana.com/answer/culture.htm |title=History and Nomenclature|access-date=2023-06-09}}</ref><ref>{{cite book|title=Geometry Connections: Mathematics for Middle School Teachers|series=Connections in mathematics courses for teachers|author1-first=John K.|author1-last=Beem|publisher=Pearson Prentice Hall|year=2006|isbn=9780131449268|page=57}}</ref>

== Etymology ==
In the ancient Greek geometry of [[Euclid's Elements|Euclid's ''Elements'']] ({{circa|300}} BC), quadrilaterals were classified into exclusive categories: [[square]]; oblong (non-square [[rectangle]]); (non-square) [[rhombus]]; rhomboid, meaning a non-rhombus non-rectangle [[parallelogram]]; or trapezium (τραπέζιον, literally "table"), meaning any quadrilateral not already included in the previous categories.<ref name=Murray>{{cite book
 | last = Murray | first = James
 | title = [[A New English Dictionary on Historical Principles]]: Founded Mainly on the Materials Collected by the Philological Society
 | volume = X
 | year = 1926
 | publisher = Clarendon Press at Oxford
 | page = 286, also see "Trapezoid", {{pgs|286–287}}
 | chapter = Trapezium
 | chapter-url = https://archive.org/details/oedxaarch/page/n298/mode/1up
}}</ref>

The Neoplatonist philosopher [[Proclus]] (mid 5th century AD) wrote an influential commentary on Euclid with a richer set of categories, which he attributed to [[Posidonius]] ({{circa|100}} BC). In this scheme, a quadrilateral can be a parallelogram or a non-parallelogram. A parallelogram can itself be a square, an oblong (non-square rectangle), a rhombus, or a rhomboid (non-rhombus non-rectangle). A non-parallelogram can be a ''trapezium'' with exactly one pair of parallel sides, which can be ''isosceles'' (with equal legs) or ''scalene'' (with unequal legs); or a ''trapezoid'' (τραπεζοειδή, literally "table-like") with no parallel sides.{{r|Murray}}<ref>{{cite book |editor-last=Morrow |editor-first=Glenn R. |title=Proclus: A commentary on the first book of Euclid's Elements |publisher=Princeton University Press |year=1970 |at=[https://archive.org/details/morrow-commentary-on-the-first-book-of-euclids-elements-en-1992/page/133/mode/2up §§&nbsp;169–174, {{pgs|133–137}}] }}</ref>

[[File:Trapezium and Trapezoid, Hutton’s mistake in 1795.png|thumb|Hutton's definitions in 1795]]
All European languages except for English follow Proclus's meanings of ''trapezium'' and ''trapezoid'',{{sfn|Conway|Burgiel|Goodman-Strauss|2016|p=[http://books.google.com/books?id=Drj1CwAAQBAJ&pg=PA286 286]}} as did English until the late 18th century, when an influential mathematical dictionary published by [[Charles Hutton]] in 1795 transposed the two terms without explanation, leading to widespread inconsistency. Hutton's change was reversed in British English in about 1875, but it has been retained in American English to the present.{{r|Murray}} Late 19th century American geometry textbooks define a trapezium as having ''no'' parallel sides, a trapezoid as having ''exactly one'' pair of parallel sides, and a parallelogram as having two sets of opposing parallel sides.{{sfn|Hopkins|1891|p=33}}{{sfn|Hobbs|1899|p=66}}
To avoid confusion between contradictory British and American meanings of ''trapezium'' and ''trapezoid'', quadrilaterals with no parallel sides have sometimes been called ''irregular quadrilaterals''.<ref>{{cite book|title=The Nature and Utility of Mathematics|author1-first=Charles|author1-last=Davies|author1-link=Charles Davies (professor)|location=New York|publisher=A.S. Barnes & Company|year=1873 |page=35}}</ref>

== Special cases ==
[[File:Trapezoid special cases.png|280px|thumb|Trapezoid special cases.  The orange figures also qualify as parallelograms.]]
An [[isosceles trapezoid]] is a trapezoid where the base angles have the same measure.{{sfn|Dodge|2012|p=82}}<ref name=posamentier-bannister>{{cite book|title=Geometry, Its Elements and Structure: Second Edition|series=Dover Books on Mathematics|author1-first=Alfred S.|author1-last=Posamentier|author2-first=Robert L.|author2-last=Bannister|edition=2nd|publisher=Courier Corporation|year=2014|isbn=9780486782164|chapter=The Trapezoid |at=§7.7, {{pgs|282–287}} }}</ref> As a consequence the two legs are also of equal length and it has [[reflection symmetry]].{{sfn|Hopkins|1891|p=34}} This is possible for acute trapezoids or right trapezoids as rectangles. An acute trapezoid is a trapezoid with two adjacent acute angles on its longer base, and the isosceles trapezoid is an example of an acute trapezoid. The isosceles trapezoid has a special case known as a three-sided trapezoid, meaning it is a trapezoid wherein two trapezoid's legs have equal lengths as the trapezoid's base at the top.{{sfn|Alsina|Nelsen|2020|p=[http://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA90 90&ndash;91]}} The isosceles trapezoid is the [[convex hull]] of an [[antiparallelogram]], a type of [[crossed quadrilateral]]. Every antiparallelogram is formed with such a trapezoid by replacing two parallel sides by the two diagonals.{{sfn|Alsina|Nelsen|2020|p=[https://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA212 212]}}

An obtuse trapezoid, on the other hand, has one acute and one obtuse angle on each base. An example is [[parallelogram]] with equal acute angles.{{sfn|Alsina|Nelsen|2020|p=[http://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA90 90&ndash;91]}}

A right trapezoid is a trapezoid with two adjacent [[right angle]]. One special type of right trapezoid is by forming three [[right triangle]]s,{{sfn|Alsina|Nelsen|2020|p=[http://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA91 91]}} which was used by [[James A. Garfield|James Garfield]] to [[Garfield's proof of the Pythagorean theorem|prove]] the [[Pythagorean theorem]].<ref>{{cite journal
 | last = Garfield |first = James | author-link = James A. Garfield
 | year = 1876
 | title = Pons Asinorum
 | jstor = 44764657
 | journal = New England Journal of Education
 | volume = 3 | issue = 14 | pages = 161
 | issn = 2578-4145
}}</ref>

A '''[[tangential trapezoid]]''' is a trapezoid that has an [[Tangential quadrilateral|incircle]].

==Condition of existence==
Four lengths ''a'', ''c'', ''b'', ''d'' can constitute the consecutive sides of a non-parallelogram trapezoid with ''a'' and ''b'' parallel only when<ref>''Ask Dr. Math'' (2008), [http://mathforum.org/library/drmath/view/72738.html "Area of Trapezoid Given Only the Side Lengths"].</ref>
:<math>\displaystyle |d-c| < |b-a| < d+c.</math>

The quadrilateral is a parallelogram when <math>d-c = b-a = 0</math>, but it is an [[ex-tangential quadrilateral]] (which is not a trapezoid) when <math>|d-c| = |b-a| \neq 0</math>.{{sfn|Josefsson|2013|p=35}}

==Characterizations==
[[File:Trapez mittellinie en labels.svg|thumb|upright=1.0|'''general trapezoid/trapezium:'''<br/>parallel sides:  <math>a,\, b </math> with  <math>a<b </math> <br/>legs:  <math> c,\, d</math><br/>diagonals:  <math>q,\, p </math><br/> midsegment:  <math>m </math><br/> height/altitude:  <math>h </math>]]
[[File:Trapez mittellinie en labels areas.svg|thumb|upright=1.0|trapezoid/trapezium with opposing triangles <math>S,\,T </math> formed by the diagonals]]
Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:
*It has two adjacent [[angle]]s that are [[supplementary angles|supplementary]], that is, they add up to 180 [[degree (angle)|degree]]s.
*The angle between a side and a [[diagonal]] is equal to the angle between the opposite side and the same diagonal.
*The diagonals cut each other in mutually the same [[ratio]] (this ratio is the same as that between the lengths of the parallel sides).
*The diagonals cut the quadrilateral into four [[Triangle|triangles]] of which one opposite pair have equal areas.{{sfn|Josefsson|2013|at=Prop. 5}}
*The product of the areas of the two triangles formed by one diagonal equals the product of the areas of the two triangles formed by the other diagonal.{{sfn|Josefsson|2013|at=Thm. 6}}
*The areas ''S'' and ''T'' of some two opposite triangles of the four triangles formed by the diagonals satisfy the equation

::<math>\sqrt{K} = \sqrt{S}+\sqrt{T},</math>

:where ''K'' is the area of the quadrilateral.{{sfn|Josefsson|2013|at=Thm. 8}}

*The midpoints of two opposite sides of the trapezoid and the intersection of the diagonals are [[collinear]].{{sfn|Josefsson|2013|at=Thm. 15}}
*The angles in the quadrilateral ''ABCD'' satisfy <math>\sin A\sin C=\sin B\sin D.</math>{{sfn|Josefsson|2013|p=25}}
*The cosines of two adjacent angles [[Addition|sum]] to 0, as do the cosines of the other two angles.{{sfn|Josefsson|2013|p=25}}
*The cotangents of two adjacent angles sum to 0, as do the cotangents of the other two adjacent angles.{{sfn|Josefsson|2013|p=26}}
*One bimedian divides the quadrilateral into two quadrilaterals of equal areas.{{sfn|Josefsson|2013|p=26}}
*Twice the length of the bimedian connecting the midpoints of two opposite sides equals the sum of the lengths of the other sides.{{sfn|Josefsson|2013|p=31}}

Additionally, the following properties are equivalent, and each implies that opposite sides ''a'' and ''b'' are parallel:
* The consecutive sides ''a'', ''c'', ''b'', ''d'' and the diagonals ''p'', ''q'' satisfy the equation{{sfn|Josefsson|2013|at=Cor. 11}}
::<math>p^2+q^2=c^2+d^2+2ab.</math>

*The distance ''v'' between the midpoints of the diagonals satisfies the equation{{sfn|Josefsson|2013|at=Thm. 12}}
::<math>v=\frac{|a-b|}{2}.</math>

== Properties ==
=== Midsegment and height ===
The ''midsegment'' or ''median'' of a trapezoid is the segment that joins the [[midpoint]]s of the legs. It is parallel to the bases. Its length ''m'' is equal to the average of the lengths of the bases ''a'' and ''b'' of the trapezoid,{{sfn|Hobbs|1899|p=58}}{{r|posamentier-bannister}}{{sfn|Dodge|2012|p=[https://books.google.com/books?id=N1b_Unptpy8C&pg=PA117 117]}}{{r|ringenberg}}
:<math>m = \frac{a + b}{2}.</math>

The midsegment of a trapezoid is one of the two [[Quadrilateral#Special line segments|bimedian]]s (the other bimedian divides the trapezoid into equal areas).

The ''height'' (or altitude) is the [[perpendicular]] distance between the bases.{{sfn|Hopkins|1891|p=33}} In the case that the two bases have different lengths (''a'' ≠ ''b''), the height of a trapezoid ''h'' can be determined by the length of its four sides using the formula<ref name="Mathworld">{{MathWorld |title=Trapezoid |urlname=Trapezoid}}</ref>
:<math>h= \frac{\sqrt{(p-2a)(p-2b)(p-2b-2d)(p-2b-2c)}}{2|b-a|}</math>

where ''c'' and ''d'' are the lengths of the legs and <math>p = a+b+c+d</math>.

=== Area ===
The area <math> K </math> of a trapezoid is given by the product of the midsegment (the average of the two bases) and the height:
<math display="block"> K = \tfrac{1}{2} (a + b) h </math>
where <math> a </math> and <math> b </math> are the lengths of the bases, and <math> h </math> is the height (the perpendicular distance between these sides).{{sfn|Dodge|2012|p=[https://books.google.com/books?id=N1b_Unptpy8C&pg=PA84 84]}} This method has been used in [[Aryabhata]]'s ''[[Aryabhatiya]]'' in section 2.8 in the classical age of Indian, yielding as a [[special case]] the well-known formula for the area of a [[triangle]], by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.

The 7th-century Indian mathematician [[Bhāskara I]] derived the following formula for the area of a trapezoid with consecutive sides <math> a </math>, <math> c </math>, <math> b </math>, <math> d </math>::
<math display="block"> K = \frac{1}{2}(a+b)\sqrt{c^2-\frac{1}{4}\left((b-a)+\frac{c^2-d^2}{b-a}\right)^2}</math>
where <math> a </math> and <math> b </math> are parallel and <math> b > a </math>.<ref>{{cite book
 | last = Puttaswamy | first = T. K.
 | url = https://books.google.com/books?id=8oVRSu692qoC&pg=PA156
 | page = 156
 | title = Mathematical Achievements of Pre-modern Indian Mathematicians
 | publisher = Elsevier
 | year = 2012
| isbn = 978-0-12-397913-1
 }}</ref> This formula can be factored into a more symmetric version<ref name=Mathworld/>
:<math>K = \frac{a+b}{4|b-a|}\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}.</math>

When one of the parallel sides has shrunk to a point (say ''a'' = 0), this formula reduces to [[Heron's formula]] for the area of a triangle.

Another equivalent formula for the area, which more closely resembles Heron's formula, is<ref name=Mathworld/>
:<math>K = \frac{a+b}{|b-a|}\sqrt{(s-b)(s-a)(s-b-c)(s-b-d)}</math>

where <math>s = \tfrac{1}{2}(a + b + c + d)</math> is the [[semiperimeter]] of the trapezoid. (This formula is similar to [[Brahmagupta's formula]], but it differs from it, in that a trapezoid might not be [[cyclic quadrilateral|cyclic]] (inscribed in a circle). The formula is also a special case of [[Bretschneider's formula]] for a general [[quadrilateral]]).

From Bretschneider's formula, it follows that
:<math>K= \sqrt{\frac{\left(ab^2-a^2 b-ad^2+bc^2\right)\left(ab^2-a^2 b-ac^2+bd^2\right)}{4(b-a)^2} - \left(\frac{c^2+d^2-a^2-b^2}{4}\right)^2}.</math>

The [[bimedian]] connecting the parallel sides bisects the area. More generally, any line drawn through the midpoint of the median parallel to the bases, that intersects the bases, bisects the area. Any triangle connecting the two ends of one leg to the midpoint of the other leg is also half of the area.{{sfn|Hopkins|1891|p=95}}

=== Diagonals ===
The lengths of the diagonals are
<math display="block"> \begin{align}
 p &= \sqrt{\frac{ab^2-a^2b-ac^2+bd^2}{b-a}}, \\
 q &= \sqrt{\frac{ab^2-a^2b-ad^2+bc^2}{b-a}},
\end{align} </math>
where <math> a </math> is the short base, <math> b </math> is the long base, and <math> c </math> and <math> d </math> are the trapezoid legs.{{sfn|Alsina|Nelsen|2020|p=[http://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA96 96]}}

If the trapezoid is divided into four triangles by its diagonals ''AC'' and ''BD'' (as shown on the right), intersecting at ''O'', then the area of {{trianglenotation|AOD}} is equal to that of {{trianglenotation|BOC}}, and the product of the areas of {{trianglenotation|AOD}} and {{trianglenotation|BOC}} is equal to that of {{trianglenotation|AOB}} and {{trianglenotation|COD}}. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.<ref name=Mathworld/>

If {{tmath|\ell}} is the length of the [[line segment]] parallel to the bases, passing through the intersection of the diagonals, with one endpoint on each leg, then {{tmath|\ell}} is the [[harmonic mean]] of the lengths of the bases:<ref>{{cite journal |title=The Harmonic Mean: A Nomograph, and some Problems |last=Skidell |first=Akiva |journal=The Mathematics Teacher |year=1977 |volume=70 |number=1 |pages=30–34 |jstor=27960699 |doi=10.5951/MT.70.1.0030 }} {{pb}} {{cite journal |last=Hoehn |first=Larry |year=1984 |title=A Geometrical Interpretation of the Weighted Mean |url=https://www.tandfonline.com/doi/abs/10.1080/00494925.1984.11972762 <!-- include a publisher URL here to work around broken DOI --> |url-access=subscription |journal=Two-Year College Mathematics Journal |volume=15 |number=2 |pages=135–139 |doi=10.1080/00494925.1984.11972762 |doi-broken-date=2 April 2025 }}</ref>
:<math>\frac{1}{\ell}=\frac{1}{2} \left( \frac{1}{a}+ \frac{1}{b} \right).</math>

The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.<ref name=Byer/>

=== Other properties ===
The center of area ([[center of mass]] for a uniform [[planar lamina|lamina]]) lies along the line segment joining the midpoints of the parallel sides, at a perpendicular distance ''x'' from the longer side ''b'' given by<ref>{{Cite web |title=Centroid, Area, Moments of Inertia, Polar Moments of Inertia, & Radius of Gyration of a General Trapezoid |url=https://www.efunda.com/math/areas/Trapezoid.cfm |access-date=2024-05-15 |website=www.efunda.com}}</ref>
:<math>x = \frac{h}{3} \left( \frac{2a+b}{a+b}\right).</math>

The center of area divides this segment in the ratio (when taken from the short to the long side)<ref name=AM>{{cite journal|last1=Apostol |first1=Tom M. |last2=Mnatsakanian |first2=Mamikon A. |title=Figures Circumscribing Circles|journal=American Mathematical Monthly|volume=111|issue=10|date=December 2004|pages=853–863|doi=10.2307/4145094|url=http://www.maa.org/sites/default/files/images/upload_library/22/Ford/Apostol853-863.pdf|access-date=2016-04-06|jstor=4145094}}</ref>{{rp|p. 862}}

:<math>\frac{a+2b}{2a+b}.</math>

If the angle bisectors to angles ''A'' and ''B'' intersect at ''P'', and the angle bisectors to angles ''C'' and ''D'' intersect at ''Q'', then<ref name=Byer>Owen Byer, Felix Lazebnik and [[Deirdre Smeltzer]], ''[{{GBurl|id=W4acIu4qZvoC|q=trapezoid}} Methods for Euclidean Geometry]'', Mathematical Association of America, 2010, p. 55.</ref>
:<math>PQ=\frac{|AD+BC-AB-CD|}{2}.</math>

== Applications ==
{{multiple image
 | image1 = Integration_num_trapezes_notation.svg
 | caption1 = The trapezoidal rule for numerical integration
 | image2 = Temple of Dendur- night.jpg
 | caption2 = The [[Temple of Dendur]] in the [[Metropolitan Museum of Art]] in [[New York City]]
 | image3 = Juanita Vilas Marchant Stenocephalidae Heteroptera HemipteraP (cropped).jpg
 | caption3 = Example of a trapeziform [[prothorax|pronotum]] outlined on a [[Stenocephalidae|spurge bug]]
 | total_width = 360
 | image4 = Ontario Highway 502.svg
 | caption4 = [[Ontario Highway 502]]
 | perrow = 2
}}
In [[calculus]], the [[definite integral]] of a [[Function of a real variable|function]] <math> f(x) </math> can be [[numerical integration|numerically approximated]] as a discrete [[summation|sum]] by [[Partition of an interval|partitioning]] the interval of integration into small uniform intervals and approximating the function's value on each interval as the average of the values at its endpoints:
<math display="block">
\int_{a}^{b} f(x)\, dx \approx \sum_{k=1}^{N} \tfrac12 \bigl( f(x_{k-1}) + f(x_{k}) \bigr) \Delta x,
</math>
where <math>N</math> is the number of intervals, <math>x_0 = a</math>, <math>x_N = b</math>, and <math>\Delta x = (b - a) / N</math>. Graphically, this amounts to approximating the region under the [[graph of a function|graph]] of the function by a collection of trapezoids, so this method is called the ''[[trapezoidal rule]]''.<ref>{{cite book
 | last1 = Varberg | first1 = Dale E.
 | last2 = Purcell | first2 = Edwin J.
 | last3 = Rigdon | first3 = Steven E.
 | title = Calculus
 | year = 2007
 | publisher = [[Pearson Prentice Hall]]
 | edition = 9th
 | isbn = 978-0131469686
 | page=264
}}</ref>

When any rectangle is viewed in [[Perspective (graphical)|perspective]] from a position which is centered on one axis but not the other, it appears to be an isosceles trapezoid, called the [[keystone effect]] because arch [[Keystone (architecture)|keystone]]s are commonly trapezoidal. For example, when a rectangular building [[façade]] is photographed from the ground at a position directly in front using a [[rectilinear lens]], the image of the building is an isosceles trapezoid. Such photographs sometimes have a "keystone transformation" applied to them to recover rectangular shapes. [[Video projector]]s sometimes apply such a keystone transformation to the recorded image before projection, so that the image projected on a flat screen appears undistorted.

[[File:14 campidoglio-altointro (1).jpg|thumb|right|upright=0.5|Piazza del Campidoglio viewed from directly above.]]
Trapezoidal doors and windows were the standard style for the [[Inca architecture|Inca]], although it can be found used by earlier cultures of the same region and did not necessarily originate with them.<ref>{{Cite web|url=http://gogeometry.com/MachuPicchu.htm|title=Machu Picchu Lost City of the Incas – Inca Geometry.|website=gogeometry.com|access-date=2018-02-13}}</ref><ref>{{cite book|title=Inka Settlement Planning|author1-first=John|author1-last=Hyslop|publisher=University of Texas Press|year=2014|isbn=9780292762640 |page=54}}</ref>
An almena, a [[battlement]] feature characteristic of [[Moorish architecture]], is trapezoidal.{{sfn|Curl|1999|p=19|loc=almena}}
Michaelangelo's redesign of the [[Piazza del Campidoglio]] (see photograph at right) incorporated a trapezoid surrounding an ellipse, giving the effect of a square surrounding a circle when seen foreshortened at ground level.{{sfn|Curl|1999|p=486|loc=Michaelangelo Buonarroti}}
[[Cinematography]] takes advantage of trapezoids in the opposite way, to produce an excessive foreshortening effect from the camera viewpoint, giving the illusion of greater depth to a room in a movie studio than the set physically has.{{sfn|Ramírez|2012|p=84}}
Trapezoids were also used to produce the visual distortions of [[Caligarism]].{{sfn|Ramírez|2012|p=84}}
[[Canal]]s and drainage [[ditch]]es commonly have a trapezoidal cross-section.

In biology, especially [[Morphology (biology)|morphology]] and [[Taxonomy (biology)|taxonomy]], terms such as ''trapezoidal'' or ''trapeziform'' commonly are useful in descriptions of particular organs or forms.<ref name="Capinera2008">{{cite book|author=John L. Capinera|title=Encyclopedia of Entomology|url={{GBurl|id=i9ITMiiohVQC|p=1247}}|date=11 August 2008|publisher=Springer Science & Business Media|isbn=978-1-4020-6242-1|pages=386, 1062, 1247}}</ref>

Trapezoids are sometimes used as a graphical symbol. In [[circuit diagram]]s, a trapezoid is the symbol for a [[multiplexer]].<ref>{{cite book
 | last = Daniels | first = Jerry
 | year = 1996
 | publisher = John Wiley & Sons
 | title = Digital Design from Zero to One
 | isbn = 978-0-471-12447-4
 | url = https://books.google.com/books?id=6DydEAAAQBAJ
 | at=[http://books.google.com/books?id=6DydEAAAQBAJ&pg=PA203 {{pgs|203}}]
}}</ref> An isosceles trapezoid is used for the shape of road signs, for example, on secondary highways in [[Ontario, Canada]].{{sfn|Alsina|Nelsen|2020|p=[https://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA93 93]}}

== Non-Euclidean geometry ==
In [[spherical geometry|spherical]] or [[hyperbolic geometry]], the internal angles of a quadrilateral do not sum to 360°, but quadrilaterals analogous to trapezoids, parallelograms, and rectangles can still be defined, and additionally there are a few new types of quadrilaterals not distinguished in the Euclidean case.

A spherical or hyperbolic trapezoid is a quadrilateral with two opposite sides, the legs, each of whose two adjacent angles sum to the same quantity; the other two sides are the bases.<ref>{{cite journal |last=Petrov |first=F. V. |journal=Matematicheskoe Prosveschenie |series=Tret’ya Seriya |volume=13 |pages=149–154 |year=2009 |script-title=ru:Вписанные четырёхугольники и трапеции в абсолютной геометрии |trans-title=Cyclic quadrilaterals and trapezoids in absolute geometry |language=ru |url=https://math.ru/lib/files/pdf/mp3/mp-13.pdf#page=149 }}</ref> As in Euclidean geometry, special cases include isosceles trapezoids whose legs are equal (as are the angles adjacent to each base), parallelograms with two pairs of opposite equal angles and two pairs of opposite equal sides, rhombuses with two pairs of opposite equal angles and four equal sides, rectangles with four equal (non-right) angles and two pairs of opposite equal sides, and squares with four equal (non-right) angles and four equal sides.

When a rectangle is cut in half along the line through the midpoints of two opposite sides, each of the resulting two pieces is an isosceles trapezoid with two right angles, called a [[Saccheri quadrilateral]]. When a rectangle is cut into quarters by the two lines through pairs of opposite midpoints, each of the resulting four pieces is a quadrilateral with three right angles called a [[Lambert quadrilateral]]. In Euclidean geometry Saccheri and Lambert quadrilaterals are merely rectangles.

== Related topics ==
[[File:Young 456 French.svg|thumb|An example of trapezoidal number: 15 = 4 + 5 + 6]]
The [[trapezoidal number]] is a set of [[positive integer]]s obtained by summing consecutively two or more positive integers greater than one, forming a trapezoidal pattern.<ref>{{cite journal
 | last1 = Gamer | first1 = Carlton
 | last2 = Roeder | first2 = David W.
 | last3 = Watkins | first3 = John J.
 | title = Trapezoidal numbers
 | doi = 10.2307/2689901
 | journal = Mathematics Magazine
 | volume = 58 | issue = 2 | year = 1985 | pages = 108–110
 | jstor = 2689901
}}</ref>

The [[crossed ladders problem]] is the problem of finding the distance between the parallel sides of a right trapezoid, given the diagonal lengths and the distance from the perpendicular leg to the diagonal intersection.{{sfnp|Alsina|Nelsen|2020|p=[http://books.google.com/books?id=CGDSDwAAQBAJ&pg=PA102 102]}}

==See also==
* [[Frustum]], a solid having trapezoidal faces
* [[Wedge (geometry)|Wedge]], a polyhedron defined by two triangles and three trapezoid faces.

== Notes ==
{{reflist}}

== Bibliography ==
* {{cite book
 | last1 = Alsina | first1 = Claudi
 | last2 = Nelsen | first2 = Roger
 | title = A Cornucopia of Quadrilaterals
 | publisher = Mathematical Association of America
 | year = 2020
 | isbn = 978-1-4704-5312-1
 | url = https://books.google.com/books?id=CGDSDwAAQBAJ
}}
* {{cite book
 | last1 = Conway | first1 = John H.
 | last2 = Burgiel | first2 = Heidi
 | last3 = Goodman-Strauss | first3 = Chaim
 | title = The Symmetries of Things
 | url = https://books.google.com/books?id=Drj1CwAAQBAJ
 | year = 2016
 | publisher = CRC Press
 | isbn = 978-1-4398-6489-0
}}
* {{cite book|title=A Dictionary of Architecture|author1-first=James Stevens|author1-last=Curl|publisher=Oxford University Press|year=1999|isbn=9780198606789}}
* {{cite book|title=Euclidean Geometry and Transformations|series=Dover Books on Mathematics|author1-first=Clayton W.|author1-last=Dodge|publisher=Courier Corporation|year=2012|isbn=9780486138428}}
* {{cite book|last=Hobbs|first=Charles Austin|title=The Elements of Plane Geometry|publisher=A. Lovell & Company|year=1899}}
* {{cite book|title=Manual of Plane Geometry|author1-first=George Irving|author1-last=Hopkins|publisher=D.C. Heath & Company|year=1891}}
* {{cite journal
 | last = Josefsson | first = Martin
 | year = 2013
 | url = http://forumgeom.fau.edu/FG2013volume13/FG201305.pdf
 | title = Characterizations of trapezoids
 | journal = Forum Geometricorum
 | volume = 13
 | pages = 23–35
| archive-url = https://web.archive.org/web/20130616225118/http://forumgeom.fau.edu/FG2013volume13/FG201305.pdf
 | archive-date = 16 June 2013
 }}
* {{cite book
 | last1 = Usiskin | first1 = Zalman
 | last2 = Griffin | first2 = Jennifer
 | pages = 49–52, 63–67
 | publisher = [[Information Age Publishing]]
 | title = The Classification of Quadrilaterals: A Study of Definition
 | year = 2008
 }}
* {{cite book|title=Architecture for the Screen: A Critical Study of Set Design in Hollywood's Golden Age|author1-first=Juan Antonio|author1-last=Ramírez|translator1-first=John F.|translator1-last=Moffitt|publisher=McFarland|year=2012|isbn=9780786469307|chapter=Architecture and Desire: The character of film constructions}}

==Further reading==
* {{cite journal
 | last1 = Fraivert | first1 = David
 | last2= Sigler | first2 = Avi
 | last3 = Stupel | first3 = Moshe
 | year = 2016
 | title = Common properties of trapezoids and convex quadrilaterals
 | journal = Journal of Mathematical Sciences: Advances and Applications
 | volume = 38
 | pages = 49–71
 | doi = 10.18642/jmsaa_7100121635
 | doi-access = free
}}

==External links==
* [https://www.encyclopediaofmath.org/index.php/Trapezium "Trapezium"] at the ''[[Encyclopedia of Mathematics]]''
* {{MathWorld |title=Right trapezoid |urlname=RightTrapezoid}}
* [http://www.mathopenref.com/trapezoid.html Trapezoid definition], [http://www.mathopenref.com/trapezoidarea.html Area of a trapezoid], [http://www.mathopenref.com/trapezoidmedian.html Median of a trapezoid] (with interactive animations)
* [http://www.elsy.at/kurse/index.php?kurs=Trapezoid+%28North+America%29&status=public Trapezoid (North America)] at elsy.at: Animated course (construction, circumference, area)
* [http://numericalmethods.eng.usf.edu/topics/trapezoidal_rule.html Trapezoidal Rule] on ''Numerical Methods for Stem Undergraduate''
* Autar Kaw and E. Eric Kalu, ''[http://www.autarkaw.com/books/numericalmethods/index.html Numerical Methods with Applications]'' (2008)

{{Polygons}}

[[Category:Elementary shapes]]
[[Category:Types of quadrilaterals]]