Cone

Measurements and equations

Volume

The volume ${\displaystyle V}$ of any conic solid, regardless of the shape of its base, is one third of the product of the area of the base ${\displaystyle A_B}$ and the height ${\displaystyle h}$

$${\displaystyle V=\frac{1}{3}{A}_{B}h.}$$

In modern mathematics, this formula can easily be computed using calculus — if ${\displaystyle A_B=k\cdot h}$, where ${\displaystyle k}$ is a coefficient, the integral

$${\displaystyle {\int }_0^{h}k{x}^{2}dx=\frac{1}{3}k{h}^3}$$

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.

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 ${\displaystyle r}$ and height ${\displaystyle h}$, the base is a circle of area ${\displaystyle \pi r^2}$ thus the formula for volume is:

$${\displaystyle V=\frac{1}{3}\pi r^{2}h}$$

Slant height

The 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 ${\displaystyle \sqrt{r^2+h^2}}$, where ${\displaystyle r}$ is the radius of the base and ${\displaystyle h}$ is the height. This can be proved by the Pythagorean theorem.

Surface area

The lateral surface area of a right circular cone is ${\displaystyle LSA=\pi r\ell }$ where ${\displaystyle r}$ is the radius of the circle at the bottom of the cone and ${\displaystyle \ell }$ is the slant height of the cone. The surface area of the bottom circle of a cone is the same as for any circle, ${\displaystyle \pi r^2}$. Thus, the total surface area of a right circular cone can be expressed as each of the following:

• Radius and height

${\displaystyle \pi r^2+\pi r\sqrt{r^2+h^2}}$

(the area of the base plus the area of the lateral surface; the term ${\displaystyle \sqrt{r^2+h^2}}$ is the slant height)

${\displaystyle \pi r\left(r+\sqrt{r^2+h^2}\right)}$

where ${\displaystyle r}$ is the radius and ${\displaystyle h}$ is the height.

• Radius and slant height

${\displaystyle \pi r^2+\pi r\ell }$

${\displaystyle \pi r(r+\ell )}$

where ${\displaystyle r}$ is the radius and ${\displaystyle \ell }$ is the slant height.

• Circumference and slant height

${\displaystyle \frac{c^2}{4\pi }+\frac{c\ell }{2}}$

${\displaystyle \left(\frac{c}{2}\right)\left(\frac{c}{2\pi }+\ell \right)}$

where ${\displaystyle c}$ is the circumference and ${\displaystyle \ell }$ is the slant height.

• Apex angle and height

${\displaystyle \pi h^2\tan \frac{\theta }{2}\left(\tan \frac{\theta }{2}+\sec \frac{\theta }{2}\right)}$

${\displaystyle -\frac{\pi h^2\sin \frac{\theta }{2}}{\sin \frac{\theta }{2}-1}}$

where ${\displaystyle \theta }$ is the apex angle and ${\displaystyle h}$ is the height.

Circular sector

The circular sector is obtained by unfolding the surface of one nappe of the cone:

• radius R

${\displaystyle R=\sqrt{r^2+h^2}}$

• arc length L

${\displaystyle L=c=2\pi r}$

• central angle φ in radians

${\displaystyle \phi =\frac{L}{R}=\frac{2\pi r}{\sqrt{r^2+h^2}}}$

Equation form

The surface of a cone can be parameterized as

${\displaystyle f(\theta ,h)=(h\cos \theta ,h\sin \theta ,h),}$

where ${\displaystyle \theta \in [0,2\pi )}$ is the angle "around" the cone, and ${\displaystyle h\in \mathbb{R}}$ is the "height" along the cone.

A right solid circular cone with height ${\displaystyle h}$ and aperture ${\displaystyle 2\theta }$, whose axis is the ${\displaystyle z}$ coordinate axis and whose apex is the origin, is described parametrically as

${\displaystyle F(s,t,u)=\left(u\tan s\cos t,u\tan s\sin t,u\right)}$

where ${\displaystyle s,t,u}$ range over ${\displaystyle [0,\theta )}$, ${\displaystyle [0,2\pi )}$, and ${\displaystyle [0,h]}$, respectively.

In implicit form, the same solid is defined by the inequalities

${\displaystyle \{F(x,y,z)\le 0,z\ge 0,z\le h\},}$

where

${\displaystyle F(x,y,z)=(x^2+y^2)(\cos \theta {)}^2-z^2(\sin \theta {)}^2.}$

More generally, a right circular cone with vertex at the origin, axis parallel to the vector ${\displaystyle d}$, and aperture ${\displaystyle 2\theta }$, is given by the implicit vector equation ${\displaystyle F(u)=0}$ where

${\displaystyle F(u)=(u\cdot d{)}^2-(d\cdot d)(u\cdot u)(\cos \theta {)}^2}$

${\displaystyle F(u)=u\cdot d-|d||u|\cos \theta }$

where ${\displaystyle u=(x,y,z)}$, and ${\displaystyle u\cdot d}$ denotes the dot product.

Projective geometry

In projective geometry, a cylinder is simply a cone whose apex is at infinity. 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 conics, which require considering the cylindrical conics.

According to G. B. Halsted, a cone is generated similarly to a Steiner conic only with a projectivity and 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'."

Generalizations

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 vector space ${\displaystyle {\mathbb{R}}^n}$ 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. In this context, the analogues of circular cones are not usually special; in fact one is often interested in polyhedral cones.

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