View source for Cone

{{short description|Geometric shape}}
{{other uses|Cone (disambiguation)}}
{{distinguish|Conical surface|Truncated dome}}
{{Infobox polyhedron
| name          = Cone
| image         = Cone with labeled Radius, Height, Angle and Side.svg|
| caption       = A right circular cone with the radius of its base ''r'', its height ''h'', its slant height ''c'' and its angle ''θ''.
| type          = Solid figure
| faces         = 1 circular face and 1 conic surface
| euler         = 2
| symmetry      = [[Orthogonal group|{{math|O(2)}}]]
| surface_area  = {{math|[[Pi|{{pi}}]]''r''<sup>2</sup> + [[Pi|{{pi}}]]''rℓ''}}
| volume        = {{math|([[Pi|{{pi}}]]''r''<sup>2</sup>''h'')/3}}
}}
[[File:Cone 3d.png|thumb|upright=1.2|A right circular cone and an oblique circular cone]]
[[File:DoubleCone.png|thumb|A double cone, not infinitely extended]]

In [[geometry]], a '''cone''' is a [[three-dimensional figure]] that tapers smoothly from a [[plane (geometry)|flat]] base (typically a [[circle]]) to a point not contained in the base, called the ''[[Apex (geometry)|apex]]'' or ''[[vertex (geometry)|vertex]]''.

A cone is formed by a set of [[line segment]]s, [[Ray (geometry)|half-line]]s, or [[Line (geometry)|line]]s connecting a common point, the apex, to all of the points on a base.  In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, it extends infinitely far. In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a ''double cone''{{anchor|Double}}. Each of the two halves of a double cone split at the apex is called a ''nappe''{{anchor|Nappe}}.

Depending on the author, the base may be restricted to a circle, any one-dimensional [[quadratic form]] in the plane, any closed [[one-dimensional space|one-dimensional figure]], or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is a [[solid geometry|solid object]]; otherwise it is an [[open surface]], a [[two-dimensional]] object in three-dimensional space. In the case of a solid object, the boundary formed by these lines or partial lines is called the ''lateral surface''; if the lateral surface is [[Unbounded set|unbounded]], it is a ''[[conical surface]]''.

The [[rotational symmetry|axis]] of a cone is the straight line passing through the apex about which the cone has a [[circular symmetry]]. {{anchor|Right circular}}In common usage in elementary geometry, cones are assumed to be ''right circular'', i.e., with a circle base [[perpendicular]] to the axis.<ref name=":1">{{Cite book|url=https://books.google.com/books?id=UyIfgBIwLMQC|title=The Mathematics Dictionary|last1=James|first1=R. C. |author-link1=Robert C. James |last2=James|first2=Glenn|date=1992-07-31|publisher=Springer Science & Business Media|isbn=9780412990410|pages=74–75}}</ref> If the cone is right circular the intersection of a plane with the lateral surface is a [[conic section]]. In general, however, the base may be any shape<ref name="grunbaum">Grünbaum, ''[[Convex Polytopes]]'', second edition, p. 23.</ref> and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite [[area (geometry)|area]], and that the apex lies outside the plane of the base). Contrasted with right cones are ''oblique cones'', in which the axis passes through the centre of the base non-perpendicularly.<ref name="MathWorld">{{MathWorld |urlname=Cone |title=Cone}}</ref> 

Depending on context, ''cone'' may refer more narrowly to either a [[convex cone]] or [[projective cone]].
Cones can be generalized to [[Dimension#Additional dimensions|higher dimensions]].

== Further terminology <span class="anchor" id="Terminology"></span>==
The perimeter of the base of a cone is called the ''directrix'', and each of the line segments between the directrix and apex is a ''generatrix'' or ''generating line'' of the lateral surface.  (For the connection between this sense of the term ''directrix'' and the [[Directrix (conic section)|directrix]] of a conic section, see [[Dandelin spheres]].)

The ''base radius'' of a circular cone is the [[radius]] of its base; often this is simply called the radius of the cone. {{anchor|Aperture}}The ''aperture'' of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle ''θ'' to the axis, the aperture is 2''θ''. In [[optics]], the angle ''θ'' is called the {{anchor |half-angle}}''half-angle'' of the cone, to distinguish it from the aperture.
[[File:Acta Eruditorum - I geometria, 1734 – BEIC 13446956.jpg|thumb|Illustration from ''Problemata mathematica...'' published in [[Acta Eruditorum]], 1734]]

[[File:Cut cone unparallel.JPG|thumb|left|A cone truncated by an inclined plane]]
A cone with a region including its apex cut off by a plane is called a ''truncated cone''; if the [[Truncation (geometry)|truncation]] plane is parallel to the cone's base, it is called a ''[[frustum]]''.<ref name=":1" />  An ''elliptical cone'' is a cone with an [[ellipse|elliptical]] base.<ref name=":1" />  A ''generalized cone'' is the surface created by the set of lines passing through a vertex and every point on a boundary (see [[Visual hull]]).

== Measurements and equations==
<!-- The formulae are correct. Please check your work before editing. --><!-- Please put proofs and derivations in [[cone (geometry) proofs]] -->

=== Volume ===
[[File:visual_proof_cone_volume.svg|thumb|[[Proof without words]] that the volume of a cone is a third of a cylinder of equal diameter and height
{|
|valign="top"|{{nowrap|1.}}||A cone and a cylinder have {{nowrap|radius ''r''}} and {{nowrap|height ''h''.}}
|-
|valign="top"|2.||The volume ratio is maintained when the height is scaled to {{nowrap|1=''h' ''= ''r'' &#8730;{{pi}}.}}
|-
|valign="top"|3.||Decompose it into thin slices.
|-
|valign="top"|4.||Using Cavalieri's principle, reshape each slice into a square of the same area.
|-
|valign="top"|5.||The pyramid is replicated twice.
|-
|valign="top"|6.||Combining them into a cube shows that the volume ratio is 1:3.
|}]]
The [[volume]] <math>V</math> of any conic solid is one third of the product of the area of the base <math>A_B</math> and the height <math>h</math><ref name=":0">{{Cite book|url=https://books.google.com/books?id=EN_KAgAAQBAJ|title=Elementary Geometry for College Students|last1=Alexander|first1=Daniel C.|last2=Koeberlein|first2=Geralyn M.|date=2014-01-01|publisher=Cengage|isbn=9781285965901}}</ref> 

<math display=block>V = \frac{1}{3}A_B h.</math>

In modern mathematics, this formula can easily be computed using calculus — it is, up to scaling, the integral

<math display="block">\int x^2 \, dx = \tfrac{1}{3} x^3</math>

Without using calculus, the formula can be proven by comparing the cone to a pyramid and applying [[Cavalieri's principle]] – specifically, comparing the cone to a (vertically scaled) right square pyramid, which forms one third of a cube. This formula cannot be proven without using such infinitesimal arguments – unlike the 2-dimensional formulae for polyhedral area, though similar to the area of the circle – and hence admitted less rigorous proofs before the advent of calculus, with the ancient Greeks using the [[method of exhaustion]]. This is essentially the content of [[Hilbert's third problem]] – more precisely, not all polyhedral pyramids are ''scissors congruent'' (can be cut apart into finite pieces and rearranged into the other), and thus volume cannot be computed purely by using a decomposition argument.<ref>{{Cite book|url=https://books.google.com/books?id=C5fSBwAAQBAJ|title=Geometry: Euclid and Beyond|last=Hartshorne|first=Robin|author-link=Robin Hartshorne|date=2013-11-11|publisher=Springer Science & Business Media|isbn=9780387226767|at=Chapter 27}}</ref>

=== Center of mass ===
The [[center of mass]] of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.

=== Right circular cone ===
====Volume====
For a circular cone with radius <math>r</math> and height <math>h</math>, the base is a circle of area <math>\pi r^2</math> thus the formula for volume is:<ref>{{Cite book|url=https://books.google.com/books?id=hMY8lbX87Y8C|title=Calculus: Single Variable|last1=Blank|first1=Brian E.|last2=Krantz|first2=Steven George|date=2006|publisher=Springer|isbn=9781931914598|at=Chapter 8}}</ref>

<math display=block>V = \frac{1}{3} \pi r^2 h </math>

====Slant height====
The [[Slant height|slant height]] of a right circular cone is the distance from any point on the [[circle]] of its base to the apex via a line segment along the surface of the cone. It is given by <math>\sqrt{r^2+h^2}</math>, where <math>r</math> is the [[radius]] of the base and <math>h</math> is the height. This can be proved by the [[Pythagorean theorem]].

====Surface area====
The [[lateral surface]] area of a right circular cone is <math>LSA = \pi r \ell</math> where <math>r</math> is the radius of the circle at the bottom of the cone and <math>\ell</math> is the slant height of the cone.<ref name=":0" />  The surface area of the bottom circle of a cone is the same as for any circle, <math>\pi r^2</math>.  Thus, the total surface area of a right circular cone can be expressed as each of the following:

*Radius and height
::<math>\pi r^2+\pi r \sqrt{r^2+h^2}</math>
:(the area of the base plus the area of the lateral surface; the term <math>\sqrt{r^2+h^2}</math> is the slant height)

::<math>\pi r \left(r + \sqrt{r^2+h^2}\right)</math>
:where <math>r</math> is the radius and <math>h</math> is the height.

[[File:Cone_surface_area.svg|thumb|Total surface area of a right circular cone, given radius 𝑟 and slant height ℓ]]
*Radius and slant height
::<math>\pi r^2+\pi r \ell</math>

::<math>\pi r(r+\ell)</math>
:where <math>r</math> is the radius and <math>\ell</math> is the slant height.

*Circumference and slant height
::<math>\frac {c^2} {4 \pi} + \frac {c\ell} 2</math>

::<math>\left(\frac c 2\right)\left(\frac c {2\pi} + \ell\right)</math>
:where <math>c</math> is the circumference and <math>\ell</math> is the slant height.

*Apex angle and height
::<math>\pi h^2 \tan \frac{\theta}{2} \left(\tan \frac{\theta}{2} + \sec \frac{\theta}{2}\right)</math>
::<math>-\frac{\pi h^2 \sin \frac{\theta}{2}}{\sin \frac{\theta}{2}-1}</math>
:where <math> \theta </math> is the apex angle and <math>h</math> is the height.

====Circular sector====
The [[circular sector]] is obtained by unfolding the surface of one nappe of the cone:

*radius ''R''
::<math>R = \sqrt{r^2+h^2}</math>

*arc length ''L''
::<math>L = c = 2\pi r</math>

*central angle ''φ'' in radians
::<math>\varphi = \frac{L}{R} = \frac{2\pi r}{\sqrt{r^2+h^2}}</math>

====Equation form====

The surface of a cone can be parameterized as
:<math>f(\theta,h) = (h \cos\theta, h \sin\theta, h ),</math>
where <math>\theta \in [0,2\pi)</math> is the angle "around" the cone, and <math>h \in \mathbb{R}</math> is the "height" along the cone.

A right solid circular cone with height <math>h</math> and aperture  <math>2\theta</math>, whose axis is the <math>z</math> coordinate axis and whose apex is the origin, is described parametrically as
:<math>F(s,t,u) = \left(u \tan s \cos t, u \tan s \sin t, u \right)</math>
where <math>s,t,u</math> range over <math>[0,\theta)</math>, <math>[0,2\pi)</math>, and <math>[0,h]</math>, respectively.

In [[Implicit function|implicit]] form, the same solid is defined by the inequalities
:<math>\{ F(x,y,z) \leq 0, z\geq 0, z\leq h\},</math>
where
:<math>F(x,y,z) = (x^2 + y^2)(\cos\theta)^2 - z^2 (\sin \theta)^2.\,</math>

More generally, a right circular cone with vertex at the origin, axis parallel to the vector <math>d</math>, and aperture <math>2\theta</math>, is given by the implicit [[vector calculus|vector]] equation <math>F(u) = 0</math> where
:<math>F(u) = (u \cdot d)^2 - (d \cdot d) (u \cdot u) (\cos \theta)^2</math>
:<math>F(u) = u \cdot d - |d| |u| \cos \theta</math>
where <math>u=(x,y,z)</math>, and <math>u \cdot d</math> denotes the [[dot product]].

=== Elliptic cone===
[[File:Elliptical Cone Quadric.Png|alt=elliptical cone quadric surface|thumb|An elliptical cone quadric surface]]
In the [[Cartesian coordinate system]], an ''elliptic cone'' is the [[Locus (mathematics)|locus]] of an equation of the form<ref>{{harvtxt |Protter |Morrey |1970 |p=583}}</ref>

<math display=block> \frac{x^2}{a^2} + \frac{y^2}{b^2} = z^2 .</math>

It is an [[Affine map|affine image]] of the right-circular ''unit cone'' with equation <math>x^2+y^2=z^2\ .</math> From the fact, that the affine image of a [[conic section]] is a conic section of the same type (ellipse, parabola,...), one gets:
*Any ''plane section'' of an elliptic cone is a conic section.
Obviously, any right circular cone contains circles. This is also true, but less obvious, in the general case (see [[circular section]]).

The intersection of an elliptic cone with a concentric sphere is a [[spherical conic]].

== Projective geometry ==
[[File:Australia Square building in George Street Sydney.jpg|thumb|upright=0.6|In [[projective geometry]], a [[Cylinder (geometry)|cylinder]] is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.]]
In [[projective geometry]], a [[cylinder (geometry)|cylinder]] is simply a cone whose apex is at infinity.<ref>{{Cite book|url=https://archive.org/details/projectivegeome04dowlgoog|title=Projective Geometry|last=Dowling|first=Linnaeus Wayland|date=1917-01-01|publisher=McGraw-Hill book Company, Incorporated|language=en}}</ref> Intuitively, if one keeps the base fixed and takes the limit as the apex goes to infinity, one obtains a cylinder, the angle of the side increasing as [[arctan]], in the limit forming a [[right angle]]. This is useful in the definition of [[degenerate conic]]s, which require considering the [[cylindrical conic]]s.

According to [[G. B. Halsted]], a cone is generated similarly to a [[Steiner conic]] only with a projectivity and [[pencil (mathematics)|axial pencils]] (not in perspective) rather than the projective ranges used for the Steiner conic:

"If two copunctual non-costraight axial pencils are projective but not perspective, the meets of correlated planes form a 'conic surface of the second order', or 'cone'."<ref>[[G. B. Halsted]] (1906) ''Synthetic Projective Geometry'', page 20</ref>

== Generalizations ==
{{Further|Hypercone}}
The definition of a cone may be extended to higher dimensions; see [[convex cone]]. In this case, one says that a [[convex set]] ''C'' in the [[real number|real]] [[vector space]] <math>\mathbb{R}^n</math> is a cone (with apex at the origin) if for every vector ''x'' in ''C'' and every nonnegative real number ''a'', the vector ''ax'' is in ''C''.<ref name="grunbaum" />  In this context, the analogues of circular cones are not usually special; in fact one is often interested in [[Convex cone#Polyhedral and finitely generated cones|polyhedral cones]].

An even more general concept is the [[topological cone]], which is defined in arbitrary topological spaces.

== See also ==
* [[Bicone]]
* [[Cone (linear algebra)]]
* [[Cylinder (geometry)]]
* [[Democritus#Mathematics|Democritus]]
* [[Generalized conic]]
* [[Hyperboloid]]
* [[List of shapes]]
* [[Pyrometric cone]]
* [[Quadric]]
* [[Rotation of axes]]
* [[Ruled surface]]
* [[Translation of axes]]

== Notes ==
{{Reflist}}

== References ==
* {{ citation | first1 = Murray H. | last1 = Protter | first2=Charles B. Jr. | last2=Morrey | year = 1970 | lccn = 76087042 | title = College Calculus with Analytic Geometry | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }}

== External links ==
{{Commons category|Cones}}
* {{MathWorld |urlname=Cone |title=Cone}}
* {{MathWorld |urlname=DoubleCone |title=Double Cone}}
* {{MathWorld |urlname=GeneralizedCone |title=Generalized Cone}}
* An interactive [http://www.mathsisfun.com/geometry/cone.html Spinning Cone] from Maths Is Fun
* [http://www.korthalsaltes.com/model.php?name_en=cone Paper model cone]
* [http://mathforum.org/library/drmath/view/55017.html Lateral surface area of an oblique cone]
* [http://www.cut-the-knot.org/Curriculum/Geometry/ConicSections.shtml Cut a Cone] An interactive demonstration of the intersection of a cone with a plane

{{Authority control}}

[[Category:Elementary shapes]]
[[Category:Surfaces]]