Pyramid (geometry)

Pyramid (geometry), solid figure formed by connecting every point on or interior to a plane polygon to a single point not in the plane (see Fig. 1 and Fig. 2). A pyramid is thus a special case of a cone or of a polyhedron, a solid bounded by planes. The polygon (in Fig. 1, ABC; in Fig. 2, DEFGH) is the base of the pyramid, and the point V (or W) is the apex or vertex; the line segments, such as VA and VB, are the lateral edges of the pyramid, and the triangular sides, such as VAB and VBC, are the lateral faces. The altitude of a pyramid is the perpendicular distance from the vertex to the plane of the base.

The name of a pyramid depends upon the shape of its base. For example, a square pyramid has a square base, while a hexagonal pyramid has a six-sided base. A triangular pyramid, Fig. 1, is also called a tetrahedron; it is bounded by four triangles, any one of which may be considered the base.

A regular pyramid has a regular polygon as the base, with the vertex perpendicular to the base at its center; the slant height of a regular pyramid is the altitude (from the vertex) of any lateral face. A frustum of a pyramid is the solid between the base and a plane parallel to the base, as in Fig. 3. A truncated pyramid is the solid between the base and a plane cutting all lateral edges, as in Fig. 4.

The lateral area of a pyramid is the sum of the areas of the lateral faces; in particular, the lateral area of a regular pyramid is sp/2, in which s is the slant height and p is the perimeter of the base. The volume of any pyramid is hK/3, in which h is the altitude of the pyramid and K is the area of the base. The volume of a pyramid is thus one-third of the volume of a prism that has the same base and altitude.