A cone is a three-dimensional geometric shape bounded by a simply connected region of a plane (the base) and a surface (the lateral surface) described by the locus of all line segments joining the perimeter of the base to a point (the apex or vertex) lying off the plane of the base. In common usage in elementary geometry, “cone” usually means a right circular cone Sometimes the term “cone” refers just to the boundary, or surface, of such a solid (also known as a conic surface). In mathematical usage, the straight lines that generate the lateral surface are often considered to be indefinitely extended in both directions; thus the cone is not delimited by a base, and extends symmetrically on both sides of the apex. Such a cone (sometimes called a double cone) consists of two halves joined at the apex, each of which is known as a nappe.

The line joining the apex and the center of the base, suitably defined, is called the axis. The perimeter of the base is called the directrix, and each of the line segments between the directrix and apex is a generatrix of the lateral surface. (The term “directrix” here should not be confused with its meaning as the generator of a conic section.) A cone with its apex cut off by a plane parallel to its base is called a truncated cone or frustum.

In general, cones may have a base of any shape, and the apex may lie anywhere outside the plane of the base. Circular cones and elliptical cones have, respectively, circular and elliptical bases. A pyramid is a special type of cone with a polygonal base. If the axis of the cone is at right angles to its base then it is said to be a right cone, otherwise it is an oblique cone.

Properties

Every conic surface is ruled and developable.

A right circular cone with a generatrix at angle θ to the axis has an aperture of 2θ.

Mathematically, an elliptical conic surface is a special case of a conic section which is formed by a “conical quadric”, which is a special case of a quadric.

A cylindrical surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction. Indeed, in projective geometry there is no difference between the cylindrical and conical surfaces, and the two halves of the latter become a single connected surface.