Cone

A (finite, circular) conical surface is a ruled surface created by fixing one end of a line segment at a point (known as the vertex or apex of the cone) and sweeping the other around the circumference of a fixed circle (known as the base). When the vertex lies above the center of the base (i.e., the angle formed by the vertex, base center, and any base radius is a right angle), the cone is known as a right cone; otherwise, the cone is termed "oblique." When the base is taken as an ellipse instead of a circle, the cone is called an elliptic cone.

In discussions of conic sections, the word "cone" is commonly taken to mean "double cone," i.e., two (possibly infinitely extending) cones placed apex to apex. The infinite double cone is a quadratic surface, and each single cone is called a "nappe." The hyperbola can then be defined as the intersection of a plane with both nappes of the double cone.

As can be seen from the above, care is needed when interpreting the unqualified term "cone" since, depending on context, it may refer to the right or oblique configurations, circular or elliptical bases, the single- or double-napped versions, the finite or infinite surface excluding the circular/elliptical base, the finite surface including it, or the finite solid bounded by the sides and base. When used without qualification, especially in elementary contexts, the term "cone" often means the filled (solid) right circular cone.

Filled (in general oblique) cones with circular base radius , base center , and vertex are represented in the Wolfram Language as Cone[x1, y1, z1, x2, y2, z2, r].

A right cone of height and base radius oriented along the -axis, with vertex pointing up, and with the base located at can be described by the parametric equations

			(1)
			(2)
			(3)

for and .

The opening angle of a right cone is the vertex angle made by a cross section through the apex and center of the base. For a cone of height and radius , it is given by

(4)

Adding the squares of (1) and (2) shows that an implicit Cartesian equation for the cone is given by

(5)

where

(6)

is the ratio of radius to height at some distance from the vertex, a quantity sometimes called the opening angle, and is the height of the apex above the plane.

The volume of a cone is

(7)

where is the base area and is the height. If the base is circular, then

			(8)
			(9)
			(10)

This amazing fact was first discovered by Eudoxus, and other proofs were subsequently found by Archimedes in On the Sphere and Cylinder (ca. 225 BC) and Euclid in Proposition XII.10 of his Elements (Dunham 1990).

The geometric centroid can be obtained by setting in the equation for the centroid of the conical frustum,

(11)

(Eshbach 1975, p. 453; Beyer 1987, p. 133) yielding

(12)

The interior of the cone of base radius , height , and mass has moment of inertia tensor about its apex of

(13)

For a right circular cone, the slant height is

(14)

and the surface area (not including the base) is

			(15)
			(16)

The locus of the apex of a variable cone containing an ellipse fixed in three-space is a hyperbola through the foci of the ellipse. In addition, the locus of the apex of a cone containing that hyperbola is the original ellipse. Furthermore, the eccentricities of the ellipse and hyperbola are reciprocals.

There are three ways in which a grid can be mapped onto a cone so that it forms a cone net (Steinhaus 1999, pp. 225-227).

The equation for a general (infinite, double-napped) cone is given by

			(17)
			(18)
			(19)

which gives coefficients of the first fundamental form

			(20)
			(21)
			(22)

second fundamental form coefficients

			(23)
			(24)
			(25)

and area element

(26)

The Gaussian curvature is

(27)

and the mean curvature is

(28)

Note that writing instead of would give a helicoid instead of a cone.