Triangular prism

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}x{}
Wythoff symbol	2 3 | 2
Coxeter-Dynkin
Symmetry	D3h, [3,2], (*33)
References	U76(a)
Dual	Triangular dipyramid
Properties	convex
Vertex figure 4.4.3

For the optical prism, see Triangular prism (optics).

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 D_3h 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:

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

Related polyhedra

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	3.∞.∞

Related polytopes

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, elongated alternated cubic honeycomb, gyrated triangular prismatic honeycomb, snub square prismatic honeycomb, triangular prismatic honeycomb, triangular-hexagonal prismatic honeycomb, truncated hexagonal prismatic honeycomb, rhombitriangular-hexagonal prismatic honeycomb, snub triangular-hexagonal prismatic honeycomb, elongated triangular prismatic honeycomb

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

tetrahedral prism	octahedral prism	cuboctahedral prism	icosahedral prism	icosidodecahedral prism	Truncated dodecahedral prism
Rhombi-cosidodecahedral prism	Rhombi-cuboctahedral prism	Truncated cubic prism	Snub dodecahedral prism	n-gonal antiprismatic prism
Cantellated 5-cell	Cantitruncated 5-cell	Runcinated 5-cell	Runcitruncated 5-cell	Cantellated tesseract	Cantitruncated tesseract	Runcinated tesseract	Runcitruncated tesseract
Cantellated 24-cell	Cantitruncated 24-cell	Runcinated 24-cell	Runcitruncated 24-cell	Cantellated 120-cell	Cantitruncated 120-cell	Runcinated 120-cell	Runcitruncated 120-cell

See also

• Prism (geometry)
• Semiregular k 21 polytope

External links

• Weisstein, Eric W., "Triangular prism" from MathWorld.
• Interactive Polyhedron: Triangular Prism