Pyramid (geometry)

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Set of pyramids
Image:Pyramid.svg
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 Surface Area
3 Pyramids with regular polygon faces
- 3.1 Symmetry
4 See also
5 External links

[edit] Volume

The volume of a pyramid is Failed to parse (Missing texvc executable; please see math/README to configure.): V = \frac{1}{3} 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.

This can be proven using calculus:

It can be proved using similarity that 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{h-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 Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{A}{h^2}(h-y)^2 .

The volume is given by the integral Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{A}{h^2} \int_0^h (h-y)^2 \, dy = \frac{-A}{3h^2} (h-y)^3 \bigg|_0^h = \frac{1}{3}Ah.

(Trivially, the volume of a square-based pyramid with an apex half the height of its base can be seen to correspond to one sixth of a cube formed by fitting six such pyramids (in opposite pairs) about a center. Since the "base times height" then corresponds to one half of the cube's volume it is therefore three times the volume of the pyramid and the factor of one-third follows.)

[edit] Surface Area

The surface area of a regular pyramid is Failed to parse (Missing texvc executable; please see math/README to configure.): A = A_b + \frac{ps}{2}

where Failed to parse (Missing texvc executable; please see math/README to configure.): 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
	Image:Tetrahedron.jpg	Image:Square pyramid.png	Image:Pentagonal pyramid.png
Class	Platonic solid	Johnson solid (J1)	Johnson solid (J2)
Base	equilaterial triangle	Square	regular pentagon
Symmetry group	T_d	C_4v	C_5v

If this were attempted with a regular hexagonal base, the equilateral triangles would have to lay flat in order to meet on the center axis, giving the pyramid zero height and zero volume (a degenerate case). With a regular polygon with more than six sides, they would not meet even then.

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. If the line connected the centroid of the base and the apex is perpendicular to the base, the pyramid is said to be a right pyramid.

[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.