Prism (geometry)

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Set of uniform prisms
(A hexagonal prism is shown)
Type	uniform polyhedron
Faces	2+n total: 2 {n} n {4}
Edges	3n
Vertices	2n
Schläfli symbol	{n}×{} or t{2, n}
Coxeter-Dynkin diagram
Vertex configuration	4.4.n
Symmetry group	D_nh, [n,2], (*n22), order 4n
Rotation group	D_n, [n,2]⁺, (n22), order 2n
Dual polyhedron	bipyramids
Properties	convex, semi-regular vertex-transitive
n-gonal prism net (n = 9 here)

In geometry, a prism is a polyhedron with a n-sided polygonal base, a translated copy (not in the same plane as the first), and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids.

General, right and uniform prisms

A right prism is a prism in which the joining edges and faces are perpendicular to the base faces. This applies if the joining faces are rectangular. If the joining edges and faces are not perpendicular to the base faces, it is called an oblique prism.

Some texts may apply the term rectangular prism or square prism to both a right rectangular-sided prism and a right square-sided prism. The term uniform prism can be used for a right prism with square sides, since such prisms are in the set of uniform polyhedra.

An n-prism, having regular polygon ends and rectangular sides, approaches a cylindrical solid as n approaches infinity.

Right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra, the other series being the antiprisms.

The dual of a right prism is a bipyramid.

A parallelepiped is a prism of which the base is a parallelogram, or equivalently a polyhedron with six faces which are all parallelograms.

A right rectangular prism is also called a cuboid, or informally a rectangular box. A right square prism is simply a square box, and may also be called a square cuboid.

An equilateral square prism is simply a cube.

Volume

The volume of a prism is the product of the area of the base and the distance between the two base faces, or the height (in the case of a non-right prism, note that this means the perpendicular distance).

The volume is therefore:

V=B\cdot h

where B is the base area and h is the height. The volume of a prism whose base is a regular n-sided polygon with side length s is therefore:

V={\frac {n}{4}}hs^{2}\cot {\frac {\pi }{n}}.

Surface area

The surface area of a right prism is 2 · B + P · h, where B is the area of the base, h the height, and P the base perimeter.

The surface area of a right prism whose base is a regular n-sided polygon with side length s and height h is therefore:

A={\frac {n}{2}}s^{2}\cot {\frac {\pi }{n}}+nsh.

Symmetry

The symmetry group of a right n-sided prism with regular base is D_nh of order 4n, except in the case of a cube, which has the larger symmetry group O_h of order 48, which has three versions of D_4h as subgroups. The rotation group is D_n of order 2n, except in the case of a cube, which has the larger symmetry group O of order 24, which has three versions of D₄ as subgroups.

The symmetry group D_nh contains inversion iff n is even.

Prismatic polytope

A prismatic polytope is a dimensional generalization of a prism. An n-dimensional prismatic polytope is constructed from two (n − 1)-dimensional polytopes, translated into the next dimension.

The prismatic n-polytope elements are doubled from the (n − 1)-polytope elements and then creating new elements from the next lower element.

Take an n-polytope with f_i i-face elements (i = 0, ..., n). Its (n + 1)-polytope prism will have 2f_i + f_i−1 i-face elements. (With f₋₁ = 0, f_n = 1.)

By dimension:

Take a polygon with n vertices, n edges. Its prism has 2n vertices, 3n edges, and 2 + n faces.
Take a polyhedron with v vertices, e edges, and f faces. Its prism has 2v vertices, 2e + v edges, 2f + e faces, and 2 + f cells.
Take a polychoron with v vertices, e edges, f faces and c cells. Its prism has 2v vertices, 2e + v edges, 2f + e faces, and 2c + f cells, and 2 + c hypercells.

Uniform prismatic polytope

A regular n-polytope represented by Schläfli symbol {p, q, ..., t} can form a uniform prismatic (n + 1)-polytope represented by a Cartesian product of two Schläfli symbols: {p, q, ..., t}×{}.