Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms. All cross-sections parallel to the base faces are the same. A prism is 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.

In the case of a rectangular or square prism there may be ambiguity because some texts may mean 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, made of regular polygons ends and rectangle 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 6 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.

Area and 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).

Symmetry
The symmetry group of a right n-sided prism with regular base is Dnh of order 4n, except in the case of a cube, which has the larger symmetry group Oh of order 48, which has three versions of D4h as subgroups. The rotation group is Dn 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 Dnh 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 2 (n-1)-dimensional polytopes, translated into the next dimension.

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

By dimension:
 * A prismatic 1-polytope is a line segment, represented by an empty Schläfli symbol {}.
 * A generalized polygonal prism is rectangle, made from two translated line segments. If it is a square it is uniform and represented as {}x{}.
 * A polyhedral prism is made from two translated polygons. A regular polygon {p} can construct the uniform polyhedral prism {p}x{}.
 * A polychoric prism is a prism made from two translated polyhedra. A regular polyhedron {p,q} can construct the uniform polychoric prism {p,q}x{}.

Higher order prismatic polytopes also exist as Cartesian products of any two polytopes. The dimension of a polytope is the product of the dimensions of the elements. The first example of these exist in 4-dimensional space and are called duoprisms as the product of two polygons. Regular duoprisms are represented as {p}x{q}.