Pyramid (geometry)

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Set of pyramids

Faces	n triangles, 1 n-gon
Edges	2n
Vertices	n+1
Symmetry group	C_nv
Dual polyhedron	Self-duals
Properties	convex

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

An n-sided pyramid is a polyhedron formed by connecting an n-sided polygonal base and a point, called the apex, by n triangular faces (n ≥ 3). In other words, it is a conic solid with polygonal base.

When unspecified, the base is usually assumed to be square. For a triangular pyramid each face can serve as base, with the opposite vertex as apex. The regular tetrahedron, one of the Platonic solids, is a triangular pyramid. The square and pentagonal pyramids can also be constructed with all faces regular, and are therefore Johnson solids. All pyramids are self-dual.

Pyramids are a subclass of the prismatoids. The 1-skeleton of pyramid is a wheel graph.

1 Volume
2 Area
3 Pyramids with regular polygon faces
- 3.1 Symmetry
4 See also
5 External links

[edit] Volume

The volume of a pyramid is $V = \frac{1}{3} Ah$ where A 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.

[edit] Area

The area of a regular pyramid is $A = A_b + \frac{ps}{2}$ where $A b$ is the area of the base, p is the perimeter of the base, and s is the slant height along the bisector of a face (ie the length from the midpoint of any edge of the base to the apex).

[edit] Pyramids with regular polygon faces

If all faces are regular polygons, the pyramid base can be a regular polygon of 3, 4 or 5 sided:

Name	Tetrahedron	Square pyramid	Pentagonal pyramid

Class	Platonic solid	Johnson solid (J1)	Johnson solid (J2)
Base	equilaterial triangle	Square	regular pentagon
Symmetry group	T_d	C_4v	C_5v

The geometric center of a square-based pyramid is located on the symmetry axis, one quarter of the way from the base to the apex.

[edit] Symmetry

If the base is regular and the apex is above the center, the symmetry group of the n-sided pyramid is C_nv of order 2n, except in the case of a regular tetrahedron, which has the larger symmetry group T_d of order 24, which has four versions of C_3v as subgroups. The rotation group is C_n of order n, except in the case of a regular tetrahedron, which has the larger rotation group T of order 12, which has four versions of C₃ as subgroups.