Prism (geometry)

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Image:Geometricprisms.gif 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 special case of a prismatoid.

A right prism is a prism in which the joining edges and faces are perpendicular to the base faces. This applies iff the joining faces are rectangular.

Prisms can be triangular, quadrilateral, rectangular, square, pentagonal (also known in optics as a pentaprism), hexagonal, etc., according to the shape of the base faces. For each we can distinguish general and right versions.

A parallelepiped is a prism of which the base is a parallelogram, or equivalently a polyhedron with 6 faces which are all parallelograms.

In the case of a rectangular or square prism there may be ambiguity because some texts may mean a right rectangular prism and a right square prism.

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.

A regular right prism is a right prism with all edges of equal length. All faces are regular polygons: two n sided ones, and n squares. For n=4 all are squares and we have the cube, which is also edge- and face-uniform and so is a Platonic solid. Note that "regular" has a weaker meaning here than in the sense of "regular polyhedron", because the only prism which is regular in that strong sense is the cube.

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 prism is a bipyramid.

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

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.