Cone

Further terminology

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 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. 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 2θ.

A cone with a region including its apex cut off by a plane is called a "truncated cone"; if the truncation plane is parallel to the cone's base, it is called a frustum. An "elliptical cone" is a cone with an elliptical base. A "generalized cone" is the surface created by the set of lines passing through a vertex and every point on a boundary (also see visual hull).

Volume

The volume V of any conic solid is one third of the product of the area of the base A B and the height h

V = 1 3 A B h .

In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral ∫ x 2 d x = 1 3 x 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.

Volume

For a circular cone with radius R and height H, the formula for volume becomes

V = ∫ 0 H r 2 π d h

where r is the radius of the cone at height h measured from the apex:

r = R h H .

Thus

V = ∫ 0 H [ R h H ] 2 π d h

and so

V = 1 3 π R 2 H .

Surface area

The lateral surface area of a right circular cone is L S A = π r l where r is the radius of the circle at the bottom of the cone and l is the lateral height (the length of a line segment from the apex of the cone along its side to its base) of the cone (given by the Pythagorean theorem l = r 2 + h 2 where h is the height of the cone). The surface area of the bottom circle of a cone is the same as for any circle, π r 2 . Thus, the total surface area of a right circular cone can be expressed as each of the following:

Radius and height

π r 2 + π r r 2 + h 2 (the area of the base plus the area of the lateral surface; the term r 2 + h 2 is the slant height) π r ( r + r 2 + h 2 ) where r is the radius and h is the height.

Radius and lateral height

π r 2 + π r l π r ( r + l ) where r is the radius and l is the lateral height.

Circumference and lateral height

c 2 4 π + c l 2 ( c 2 ) ( c 2 π + l ) where c is the circumference and l is the lateral height.

Apex angle and height

π h 2 tan ⁡ Θ 2 ( tan ⁡ Θ 2 + sec ⁡ Θ 2 ) where Θ is the apex angle and h is the height.

Equation form

A right solid circular cone with height h and aperture 2 θ , whose axis is the z coordinate axis and whose apex is the origin, is described parametrically as

F ( s , t , u ) = ( u tan ⁡ s cos ⁡ t , u tan ⁡ s sin ⁡ t , u )

where s , t , u range over [ 0 , θ ) , [ 0 , 2 π ) , and [ 0 , h ] , respectively.

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

{ F ( x , y , z ) ≤ 0 , z ≥ 0 , z ≤ h } ,

where

F ( x , y , z ) = ( x 2 + y 2 ) ( cos ⁡ θ ) 2 − z 2 ( sin ⁡ θ ) 2 .

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

F ( u ) = ( u ⋅ d ) 2 − ( d ⋅ d ) ( u ⋅ u ) ( cos ⁡ θ ) 2   or   F ( u ) = u ⋅ d − | d | | u | cos ⁡ θ

where u = ( x , y , z ) , and u ⋅ d denotes the dot product.

Elliptic cone

In the Cartesian coordinate system, an elliptic cone is the locus of an equation of the form

x 2 a 2 + y 2 b 2 = z 2 c 2 .

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.

Higher dimensions

The definition of a cone may be extended to higher dimensions (see convex cones). In this case, one says that a convex set C in the real vector space Rⁿ 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.