Editing Cone (section)

== 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]].