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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} 
CoxeterDynkin 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, semiregular vertextransitive 
ngonal prism net (n = 9 here) 
In geometry, a prism is a polyhedron with an nsided 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 crosssections 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.
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 rectangularsided prism and a right squaresided 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 nprism, 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.
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 nonright prism, note that this means the perpendicular distance).
The volume is therefore:
where B is the base area and h is the height. The volume of a prism whose base is a regular nsided polygon with side length s is therefore:
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 nsided polygon with side length s and height h is therefore:
The symmetry group of a right nsided 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 D_{4} as subgroups.
The symmetry group D_{nh} contains inversion iff n is even.
A prismatic polytope is a higherdimensional generalization of a prism. An ndimensional prismatic polytope is constructed from two (n − 1)dimensional polytopes, translated into the next dimension.
The prismatic npolytope elements are doubled from the (n − 1)polytope elements and then creating new elements from the next lower element.
Take an npolytope with f_{i} iface elements (i = 0, ..., n). Its (n + 1)polytope prism will have 2f_{i} + f_{i−1} iface elements. (With f_{−1} = 0, f_{n} = 1.)
By dimension:
A regular npolytope 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}×{}.
By dimension:
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 4dimensional space are called duoprisms as the product of two polygons. Regular duoprisms are represented as {p}×{q}.
Symmetry  3  4  5  6  7  8  9  10  11  12 

[2n,2] [n,2] [2n,2^{+}]  
Image  
As spherical polyhedra  
Image 
