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. These joining faces are not only parallelograms, but also rectangles. All cross-sections parallel to the base faces are the same. A prism is a subclass of the prismatoids.
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.

[edit] 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).

[edit] 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.

[edit] 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.

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 fi i-face elements (i=0..n). Its (n+1)-polytope prism will have 2*fi+fi-1 i-face elements. (With f-1=0, fn=1.)

By dimension:

* Take a polygon with n vertices, n edges. Its prism has 2n vertices, 2n+n 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.