In the pure
mathematical sense, a
three-dimensional solid created by extruding any
two-dimensional primitive (called the
base) along a
straight line segment, linearly tapering off to a
point at the end of the line segment. The
area is defined as
al/3, where
a is the area of the base and
l is the length of the line segment as projected onto the base's surface normal (i.e. the
height of the cone if the base were to lie flat; this can be mathematically computed as l=N(L
dot N)/(||L||*||N|| where L is the line segment and N is the surface normal of the base). The line is typically not in the base's plane.
Used in the general sense (and, regrettably, even in mathematics), the base is usually a circle, and the line segment is parallel to the normal. There is also a special branch of geometry devoted to the study of the intersection of a plane with an infinite cone (one which extends infinitely in both directions - after it tapers to a point, it then begins to expand at the same rate, forming a pair of infinitely-long circle-based cones).
A true cone with the point truncated by intersection with a half-plane is called a frustum.
Note that a mathematical pyramid is a class of true cone.
Oh, and through integration, it's a simple matter to prove that the area is always al/3 regardless of the shape of the base.