Pyramid (geometry)

Set of pyramids
Faces	k triangles, 1 n-gon
Edges	2n
Vertices	n + 1
Symmetry group	Cnv, [n], (*nn)
Dual polyhedron	Self-dual
Properties	convex

The 1-skeleton of pyramid is a wheel graph

This article is about the polyhedron pyramid (a 3-dimensional shape). For other versions including architectural pyramids, see Pyramid (disambiguation).

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base.

A pyramid with an n-sided base will have n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual.

When unspecified, the base is usually assumed to be square.

If the base is a regular polygon and the apex is above the center of the polygon, an n-gonal pyramid will have C_nv symmetry.

Pyramids are a subclass of the prismatoids.

Pyramids with regular polygon faces

The regular tetrahedron, one of the Platonic solids, is a triangular pyramid all of whose faces are equilateral triangles. Besides the triangular pyramid, only the square and pentagonal pyramids can be composed of regular convex polygons, in which case they are Johnson solids.

Tetrahedron	Square pyramid	Pentagonal pyramid

Star pyramids

Pyramids with regular star polygon bases are called star pyramids. For example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides.

Volume

See also: Cone (geometry) – Volume

The volume of a pyramid is $\scriptstyle{V=} \tfrac{1}{3}\scriptstyle{Bh}$ where B is the area of the base and h the height from the base to the apex. This works for any location of the apex, provided that h is measured as the perpendicular distance from the plane which contains the base. In 499 AD Aryabhata, a mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya (section 2.6).

The formula can be formally proved using calculus: By similarity, the dimensions of a cross section parallel to the base increase linearly from the apex to the base. Then, the cross section at any height y is the base scaled by a factor of $\scriptstyle{1 -} \tfrac{y}{h}$, where h is the height from the base to the apex. Since the area of any shape is multiplied by the square of the shape's scaling factor, the area of a cross section at height y is $\tfrac{B(h - y)^2}{h^2}$. The volume is given by the integral

$\frac{B}{h^2} \int_0^h (h-y)^2 \, dy = \frac{-B}{3h^2} (h-y)^3 \bigg|_0^h = \tfrac{1}{3}Bh.$

The volume of a pyramid whose base is an n-sided regular polygon with side length s and whose height is h is therefore:

$V = \frac{n}{12}hs^2 \cot\frac{\pi}{n}.$

The volume of a pyramid whose base is a regular n-sided polygon with radius R is therefore:

$V = \frac{nR^2h}{6} \sin{\frac{2\pi}{n}}.$

The same equation, $\scriptstyle{V=} \tfrac{1}{3}\scriptstyle{Bh}$, also holds for cones with any base (not necessarily a polyhedron). This can be proven by the same argument as above, or, for sufficiently simple cones, by approximating the cone by pyramids; see volume of a cone.

Surface area

The surface area of a pyramid is $\scriptstyle{A= B +} \tfrac{PL}{2}$ where B is the base area, P is the base perimeter and L is the slant height $\scriptstyle{L=} \sqrt{\scriptstyle{h^2+r^2}}$ where h is the pyramid altitude and r is the inradius of the base.

See also

• Bipyramid
• Cone (geometry)
• Trigonal pyramid (chemistry)
• Frustum

External links

• Weisstein, Eric W., "Pyramid" from MathWorld.
• Olshevsky, George, Pyramid at Glossary for Hyperspace.
• The Uniform Polyhedra