Pyramid

A pyramid is a polyhedron with one face (known as the "base") a polygon and all the other faces triangles meeting at a common polygon vertex (known as the "apex"). A right pyramid is a pyramid for which the line joining the centroid of the base and the apex is perpendicular to the base. A regular pyramid is a right pyramid whose base is a regular polygon. An -gonal regular pyramid (denoted ) having equilateral triangles as sides is possible only for , 4, 5. These correspond to the regular tetrahedron, square pyramid, and pentagonal pyramid, respectively.

Canonical -pyramids are illustrated above for to 7.

The illustration above shows canonical -pyramids together with their duals. As can be seen, such pyramids are self-dual, corresponding to the fact that a pyramid's skeleton (a wheel graph) is a self-dual graph. Canonical -pyramids with unit midradius and midcenter at the origin have regular polygon base circumradius

(1)

and base and apex at heights

			(2)
			(3)

giving an overall height

(4)

The corresponding edges lengths, generalized diameter, circumradius, surface area, and volume are

			(5)
			(6)
			(7)
			(8)

Nets for the canonical -dipyramids for , 4, ..., 10 are illustrated above. The faces of the canonical -pyramid are isosceles triangles with angles

			(9)
			(10)

An arbitrary pyramid has a single cross-sectional shape whose lengths scale linearly with height. Therefore, the area of a cross section scales quadratically with height, decreasing from at the base () to 0 at the apex (assumed to lie at a height ). The area at a height above the base is therefore given by

(11)

As a result, the volume of a pyramid, regardless of base shape or position of the apex relative to the base, is given by

			(12)
			(13)
			(14)

Note that this formula also holds for the cone, elliptic cone, etc.

The volume of a pyramid whose base is a regular -sided polygon with side is therefore

(15)

Expressing in terms of the circumradius of the base gives

(16)

(Lo Bello 1988, Gearhart and Schulz 1990).

The geometric centroid is the same as for the cone, given by

(17)

The lateral surface area of a pyramid is

(18)

where is the slant height and is the base perimeter.

Joining two pyramids together at their bases gives a dipyramid, also called a bipyramid.