Cone

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. In optics, the angle θ is called the half-angle of the cone, to distinguish it from the aperture. , 1734 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 (see Visual hull). == Measurements and equations==

Volume that the volume of a cone is a third of a cylinder of equal diameter and height The volume V of any conic solid, regardless of the shape of its base, is one third of the product of the area of the base A_B and the height h V = \frac{1}{3}A_B h. In modern mathematics, this formula can easily be computed using calculus — if A_B = k \cdot h, where k is a coefficient, the integral \int_0^{h} k x^2 \, dx = \tfrac{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 r and height h, the base is a circle of area \pi r^2 thus the formula for volume is: 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 \sqrt{r^2+h^2}, where r is the radius of the base and h is the height. This can be proved by the Pythagorean theorem. Surface area The lateral surface area of a right circular cone is LSA = \pi r \ell where r is the radius of the circle at the bottom of the cone and \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, \pi r^2. Thus, the total surface area of a right circular cone can be expressed as each of the following: • Radius and height ::\pi r^2+\pi r \sqrt{r^2+h^2} :(the area of the base plus the area of the lateral surface; the term \sqrt{r^2+h^2} is the slant height) ::\pi r \left(r + \sqrt{r^2+h^2}\right) :where r is the radius and h is the height. • Radius and slant height ::\pi r^2+\pi r \ell ::\pi r(r+\ell) :where r is the radius and \ell is the slant height. • Circumference and slant height ::\frac {c^2} {4 \pi} + \frac {c\ell} 2 ::\left(\frac c 2\right)\left(\frac c {2\pi} + \ell\right) :where c is the circumference and \ell is the slant height. • Apex angle and height ::\pi h^2 \tan \frac{\theta}{2} \left(\tan \frac{\theta}{2} + \sec \frac{\theta}{2}\right) ::-\frac{\pi h^2 \sin \frac{\theta}{2}}{\sin \frac{\theta}{2}-1} :where \theta is the apex angle and h is the height. Circular sector The circular sector is obtained by unfolding the surface of one nappe of the cone: • radius R ::R = \sqrt{r^2+h^2} • arc length L ::L = c = 2\pi r • central angle φ in radians ::\varphi = \frac{L}{R} = \frac{2\pi r}{\sqrt{r^2+h^2}} Equation form The surface of a cone can be parameterized as :f(\theta,h) = (h \cos\theta, h \sin\theta, h ), where \theta \in [0,2\pi) is the angle "around" the cone, and h \in \mathbb{R} is the "height" along the cone. A right solid circular cone with height h and aperture 2\theta, whose axis is the z coordinate axis and whose apex is the origin, is described parametrically as :F(s,t,u) = \left(u \tan s \cos t, u \tan s \sin t, u \right) where s,t,u range over [0,\theta), [0,2\pi), and [0,h], respectively. In implicit form, the same solid is defined by the inequalities :\{ F(x,y,z) \leq 0, z\geq 0, z\leq h\}, where :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 d, and aperture 2\theta, is given by the implicit vector equation F(u) = 0 where :F(u) = (u \cdot d)^2 - (d \cdot d) (u \cdot u) (\cos \theta)^2 :F(u) = u \cdot d - |d| |u| \cos \theta where u=(x,y,z), and u \cdot d denotes the dot product. == Projective geometry ==

, a 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 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 \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. == See also ==