Triangular Prism
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Triangular Prism
Uniform triangular prism
Type Prismatic uniform polyhedron
Elements F = 5, E = 9
V = 6 (? = 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
3D model of a (uniform) triangular prism

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. A right triangular prism has rectangular sides, otherwise it is oblique. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides.

Equivalently, it is a polyhedron 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, 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:

${\displaystyle V={\frac {bhl}{2}},}$

where b is the length of one side of the triangle, h is the length of an altitude drawn to that side, and l is the distance between the triangular faces.

## Truncated triangular prism

A truncated right triangular prism has one triangular face truncated (planed) at an oblique angle.[1]

The volume of a truncated triangular prism with base area A and the three heights h1, h2, and h3 is determined by[2]

${\displaystyle V={\frac {A(h_{1}+h_{2}+h_{3})}{3}}.}$

## Facetings

There are two full D2h symmetry facetings of a triangular prism, both with 6 isosceles triangle faces, one keeping the original top and bottom triangles, and one the original squares. Two lower C3v symmetry faceting have one base triangle, 3 lateral crossed square faces, and 3 isosceles triangle lateral faces.

Convex Facetings
D3h symmetry C3v symmetry
2 {3}
3 {4}
3 {4}
6 ( ) v { }
2 {3}
6 ( ) v { }
1 {3}
3 t'{2}
6 ( ) v { }
1 {3}
3 t'{2}
3 ( ) v { }

## Related polyhedra and tilings

A regular tetrahedron or tetragonal disphenoid can be dissected into two halves with a central square. Each half is a topological triangular prism.
Family of uniform n-gonal 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 ...
Family of convex cupolae
n 2 3 4 5 6
Name {2} || t{2} {3} || t{3} {4} || t{4} {5} || t{5} {6} || t{6}
shortcode 2c 3c 4c 5c 6c
Cupola
Digonal cupola

Triangular cupola

Square cupola

Pentagonal cupola

Hexagonal cupola
(Flat)
Related
uniform
polyhedra
Triangular prism
Cubocta-
hedron

Rhombi-
cubocta-
hedron

Rhomb-
icosidodeca-
hedron

Rhombi-
trihexagonal
tiling

### Symmetry mutations

This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.

*n32 symmetry mutation of truncated tilings: t{n,3}
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
[12i,3] [9i,3] [6i,3]
Truncated
figures
Symbol t{2,3} t{3,3} t{4,3} t{5,3} t{6,3} t{7,3} t{8,3} t{∞,3} t{12i,3} t{9i,3} t{6i,3}
Triakis
figures
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.?.?

This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.

This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.

*n32 symmetry mutation of expanded tilings: 3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paracomp.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Figure
Config. 3.4.2.4 3.4.3.4 3.4.4.4 3.4.5.4 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4

### Compounds

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.

### Honeycombs

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

### Related polytopes

The triangular prism is first in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (equilateral triangles and squares in the case of the triangular prism). In Coxeter's notation the triangular prism is given the symbol −121.

k21 figures in n dimensional
Space Finite Euclidean Hyperbolic
En 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+ E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8++
Coxeter
diagram
Symmetry [3-1,2,1] [30,2,1] [31,2,1] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 1,920 51,840 2,903,040 696,729,600
Graph - -
Name −121 021 121 221 321 421 521 621

### Four dimensional space

The triangular prism exists as cells of a number of four-dimensional uniform 4-polytopes, including:

## References

1. ^ Kern, William F.; Bland, James R. (1938). Solid Mensuration with proofs. p. 81. OCLC 1035479.
2. ^ "Volume of truncated prism". Mathematics Stack Exchange. Retrieved 2019.

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