Set of uniform ngonal prisms  

Type  uniform in the sense of semiregular polyhedron 
Faces 

Edges  3n 
Vertices  2n 
Vertex configuration  4.4.n 
Schläfli symbol  {n}×{ } ^{[1]} t{2,n} 
Conway notation  Pn 
Coxeter diagram  
Symmetry group  D_{nh}, [n,2], (*n22), order 4n 
Rotation group  D_{n}, [n,2]^{+}, (n22), order 2n 
Dual polyhedron  convex dualuniform ngonal bipyramid 
Properties  convex, regular polygon faces, isogonal, translated bases, sides ⊥ bases 
Net  
Example: net of uniform enneagonal prism (n = 9) 
In geometry, a prism is a polyhedron comprising an nsided polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All crosssections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.
Like many basic geometric terms, the word prism (from Greek πρίσμα (prisma) 'something sawed') was first used in Euclid's Elements. Euclid defined the term in Book XI as “a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms”. However, this definition has been criticized for not being specific enough in relation to the nature of the bases, which caused confusion among later geometry writers.^{[2]}^{[3]}
An oblique prism is a prism in which the joining edges and faces are not perpendicular to the base faces.
Example: a parallelepiped is an oblique prism whose base is a parallelogram, or equivalently a polyhedron with six parallelogram faces.
A right prism is a prism in which the joining edges and faces are perpendicular to the base faces.^{[4]} This applies if and only if all the joining faces are rectangular.
The dual of a right nprism is a right nbipyramid.
A right prism (with rectangular sides) with regular ngon bases has Schläfli symbol { }×{n}. It approaches a cylinder as n approaches infinity.
Note: some texts may apply the term rectangular prism or square prism to both a right rectangularbased prism and a right squarebased prism.
A regular prism is a prism with regular bases.
A uniform prism or semiregular prism is a right prism with regular bases and all edges of the same length.
Thus all the side faces of a uniform prism are squares.
Thus all the faces of a uniform prism are regular polygons. Also, such prisms are isogonal; thus they are uniform polyhedra alright. They form one of the two infinite series of semiregular polyhedra, the other series being formed by the antiprisms.
A uniform ngonal prism has Schläfli symbol t{2,n}.
Family of uniform ngonal prisms  

Prism name  Digonal prism  (Trigonal) Triangular prism 
(Tetragonal) Square prism 
Pentagonal prism  Hexagonal prism  Heptagonal prism  Octagonal prism  Enneagonal prism  Decagonal prism  Hendecagonal prism  Dodecagonal prism  ...  Apeirogonal prism 
Polyhedron image  ...  
Spherical tiling image  Plane tiling image  
Vertex config.  2.4.4  3.4.4  4.4.4  5.4.4  6.4.4  7.4.4  8.4.4  9.4.4  10.4.4  11.4.4  12.4.4  ...  ∞.4.4 
Coxeter diagram  ... 
The volume of a prism is the product of the area of the base by the height, i.e. the distance between the two base faces (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 an nsided regular polygon with side length s is therefore:
The surface area of a right prism is:
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 with height h, is therefore:
P3 
P4 
P5 
P6 
P7 
P8 
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.
The hosohedra and dihedra also possess dihedral symmetry, and an ngonal prism can be constructed via the geometrical truncation of an ngonal hosohedron, as well as through the cantellation or expansion of an ngonal dihedron.
A truncated prism is a prism with nonparallel top and bottom faces.^{[5]}
A twisted prism is a nonconvex polyhedron constructed from a uniform nprism with each side face bisected on the square diagonal, by twisting the top, usually by π/n radians (180/n degrees) in the same direction, causing sides to be concave.^{[6]}^{[7]}
A twisted prism cannot be dissected into tetrahedra without adding new vertices. The smallest case: the triangular form, is called a Schönhardt polyhedron.
An ngonal twisted prism is topologically identical to the ngonal uniform antiprism, but has half the symmetry group: D_{n}, [n,2]^{+}, order 2n. It can be seen as a nonconvex antiprism, with tetrahedra removed between pairs of triangles.
3gonal  4gonal  12gonal  

Schönhardt polyhedron 
Twisted square prism 
Square antiprism 
Twisted dodecagonal antiprism 
A frustum is a similar construction to a prism, with trapezoid lateral faces and differently sized top and bottom polygons.
Further information: Prismatic uniform polyhedron 
A star prism is a nonconvex polyhedron constructed by two identical star polygon faces on the top and bottom, being parallel and offset by a distance and connected by rectangular faces. A uniform star prism will have Schläfli symbol {p/q} × { }, with p rectangle and 2 {p/q} faces. It is topologically identical to a pgonal prism.
{ }×{ }_{180}×{ }  t_{a}{3}×{ }  {5/2}×{ }  {7/2}×{ }  {7/3}×{ }  {8/3}×{ }  

D_{2h}, order 8  D_{3h}, order 12  D_{5h}, order 20  D_{7h}, order 28  D_{8h}, order 32  
A crossed prism is a nonconvex polyhedron constructed from a prism, where the vertices of one base are inverted around the center of this base (or rotated by 180°). This transforms the side rectangular faces into crossed rectangles. For a regular polygon base, the appearance is an ngonal hour glass. All oblique edges pass through a single body center. Note: no vertex is at this body centre. A crossed prism is topologically identical to an ngonal prism.
{ }×{ }_{180}×{ }_{180}  t_{a}{3}×{ }_{180}  {3}×{ }_{180}  {4}×{ }_{180}  {5}×{ }_{180}  {5/2}×{ }_{180}  {6}×{ }_{180}  

D_{2h}, order 8  D_{3d}, order 12  D_{4h}, order 16  D_{5d}, order 20  D_{6d}, order 24  
A toroidal prism is a nonconvex polyhedron like a crossed prism, but without bottom and top base faces, and with simple rectangular side faces closing the polyhedron. This can only be done for evensided base polygons. These are topological tori, with Euler characteristic of zero. The topological polyhedral net can be cut from two rows of a square tiling (with vertex configuration 4.4.4.4): a band of n squares, each attached to a crossed rectangle. An ngonal toroidal prism has 2n vertices, 2n faces: n squares and n crossed rectangles, and 4n edges. It is topologically selfdual.
D_{4h}, order 16  D_{6h}, order 24 
v=8, e=16, f=8  v=12, e=24, f=12 
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:
See also: Uniform 4polytope § Prismatic_uniform 4polytopes 
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 or more polytopes. The dimension of a product polytope is the sum of the dimensions of its elements. The first examples of these exist in 4dimensional space; they are called duoprisms as the product of two polygons in 4dimensions.
Regular duoprisms are represented as {p}×{q}, with pq vertices, 2pq edges, pq square faces, p qgon faces, q pgon faces, and bounded by p qgonal prisms and q pgonal prisms.
For example, {4}×{4}, a 44 duoprism is a lower symmetry form of a tesseract, as is {4,3}×{ }, a cubic prism. {4}×{4}×{ } (44 duoprism prism), {4,3}×{4} (cube4 duoprism) and {4,3,3}×{ } (tesseractic prism) are lower symmetry forms of a 5cube.