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Uniform triangular prism
Type Prismatic uniform polyhedron
Elements F = 5, E = 9
V = 6 (χ = 2)
Faces by sides 3{4}+2{3}
Schläfli symbol t{2,3} or {3}×{}
Wythoff symbol 2 3 | 2
Coxeter diagram
Symmetry group D3h, [3,2], (*322), order 12
Rotation group D3, [3,2]+, (322), order 6
References U76(a)
Dual Triangular dipyramid
Properties convex

Vertex figure
4.4.3

In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides.

Equivalently, it is a pentahedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are parallelograms. All cross-sections parallel to the base faces are the same triangle.

## As a semiregular (or uniform) polyhedron

A right triangular prism is semiregular or, more generally, a uniform polyhedron if the base faces are equilateral triangles, and the other three faces are squares. It can be seen as a truncated trigonal hosohedron, represented by Schläfli symbol t{2,3}. Alternately it can be seen as the Cartesian product of a triangle and a line segment, and represented by the product {3}x{}. The dual of a triangular prism is a triangular bipyramid.

The symmetry group of a right 3-sided prism with triangular base is D3h of order 12. The rotation group is D3 of order 6. The symmetry group does not contain inversion.

## Volume

The volume of any prism is the product of the area of the base and the distance between the two bases. In this case the base is a triangle so we simply need to compute the area of the triangle and multiply this by the length of the prism:

${\displaystyle V={\frac {1}{2))bhl}$ where b is the triangle base length, h is the triangle height, and l is the length between the triangles.

It is related to the following sequence of uniform truncated polyhedra and tilings with 3.2n.2n a vertex configuration and [n,3] Coxeter group symmetry.

 3.4.4 3.6.6 3.8.8 3.10.10 3.12.12 3.14.14 3.16.16 File:Hyperbolic tiling o3x∞x.png3.∞.∞

There are 4 uniform compounds of triangular prisms:

Compound of four triangular prisms, Compound of eight triangular prisms, Compound of ten triangular prisms, Compound of twenty triangular prisms

There are 9 uniform honeycombs that include triangular prism cells:

[Gyroelongated alternated cubic honeycomb]],

The triangular prism exists as cells of a number of four-dimensional uniform polychora, including: