Triangular prism

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Uniform Triangular prism
Triangular prism
TypePrismatic uniform polyhedron
ElementsF = 5, E = 9
V = 6 (χ = 2)
Faces by sides3{4}+2{3}
Schläfli symbolt{2,3} or {3}x{}
Wythoff symbol2 3 | 2
Coxeter-DynkinCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png
Symmetry groupD3h, [3,2], (*322), order 12
Rotation groupD3, [3,2]+, (322), order 6
ReferencesU76(a)
DualTriangular dipyramid
Propertiesconvex
Triangular prism
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[edit]

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[edit]

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 and tilings[edit]

Family of uniform prisms
Symmetry3456789101112
[2n,2]
[n,2]
[2n,2+]
[2n+,2]
CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 8.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 9.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 10.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 10.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 10.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 11.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 12.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 12.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node h.pngCDel 12.pngCDel node h.pngCDel 2.pngCDel node 1.png
ImageTriangular prism.pngTetragonal prism.pngPentagonal prism.pngHexagonal prism.pngPrism 7.pngOctagonal prism.pngPrism 9.pngDecagonal prism.pngHendecagonal prism.pngDodecagonal prism.png
As spherical polyhedra
ImageSpherical triangular prism.pngSpherical square prism.png
Spherical square prism2.png
Spherical pentagonal prism.pngSpherical hexagonal prism.png
Spherical hexagonal prism2.png
Spherical heptagonal prism.pngSpherical octagonal prism.png
Spherical octagonal prism2.png
Spherical decagonal prism.png
Spherical decagonal prism2.png
Family of cupolae
23456
Triangular prism wedge.png
Digonal cupola
Triangular cupola.png
Triangular cupola
Square cupola.png
Square cupola
Pentagonal cupola.png
Pentagonal cupola
Hexagonal cupola flat.png
Hexagonal cupola
(Flat)

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.

Dimensional family of truncated polyhedra and tilings: 3.2n.2n
Symmetry
*n32
[n,3]
SphericalEuclideanHyperbolic...
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
 
*832
[8,3]...
 
*∞32
[∞,3]
 
Truncated
figures
Spherical triangular prism.png
3.4.4
Uniform tiling 332-t01-1-.png
3.6.6
Uniform tiling 432-t01.png
3.8.8
Uniform tiling 532-t01.png
3.10.10
Uniform tiling 63-t01.png
3.12.12
Uniform tiling 73-t01.png
3.14.14
Uniform tiling 83-t01.png
3.16.16
H2 tiling 23i-3.png
3.∞.∞
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png
t{2,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t{3,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
t{4,3}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
t{5,3}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
t{6,3}
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png
t{7,3}
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png
t{8,3}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png
t{∞,3}
Uniform dual figures
Triakis
figures
Triangular dipyramid.png
V3.4.4
Triakistetrahedron.jpg
V3.6.6
Triakisoctahedron.jpg
V3.8.8
Triakisicosahedron.jpg
V3.10.10
Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg
V3.12.12
Ord7 triakis triang til.png
V3.14.14
Ord8 triakis triang til.png
V3.16.16
Ord-infin triakis triang til.png
V3.∞.∞
CoxeterCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.png

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.

Dimensional family of expanded polyhedra and tilings: 3.4.n.4
Symmetry
*n32
[n,3]
SphericalEuclideanHyperbolic...
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
 
*832
[8,3]...
 
*∞32
[∞,3]
 
Expanded
figure
Spherical triangular prism.png
3.4.2.4
Uniform tiling 332-t02.png
3.4.3.4
Uniform tiling 432-t02.png
3.4.4.4
Uniform tiling 532-t02.png
3.4.5.4
Uniform polyhedron-63-t02.png
3.4.6.4
Uniform tiling 73-t02.png
3.4.7.4
Uniform tiling 83-t02.png
3.4.8.4
H2 tiling 23i-5.png
3.4.∞.4
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{2,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{4,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{5,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{6,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{7,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{8,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{∞,3}
Deltoidal figureTriangular dipyramid.png
V3.4.2.4
Rhombicdodecahedron.jpg
V3.4.3.4
Deltoidalicositetrahedron.jpg
V3.4.4.4
Deltoidalhexecontahedron.jpg
V3.4.5.4
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
V3.4.6.4
Deltoidal triheptagonal til.png
V3.4.7.4
Deltoidal trioctagonal til.png
V3.4.8.4
Deltoidal triapeirogonal til.png
V3.4.∞.4
CoxeterCDel node f1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.png

Compounds[edit]

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[edit]

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[edit]

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
En345678910
Coxeter
group
E3=A2×A1E4=A4E5=D5E6E7E8E9 = {\tilde{E}}_{8} = E8+E10 = E8++
Coxeter
diagram
CDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Symmetry
(order)
[3-1,2,1]
(12)
[30,2,1]
(120)
[31,2,1]
(192)
[32,2,1]
(51,840)
[33,2,1]
(2,903,040)
[34,2,1]
(696,729,600)
[35,2,1]
(∞)
[36,2,1]
(∞)
GraphTriangular prism.png4-simplex t1.svgDemipenteract graph ortho.svgE6 graph.svgE7 graph.svgE8 graph.svg
Name−121021121221321421521621

Four dimensional space[edit]

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

tetrahedral prism
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png
octahedral prism
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
cuboctahedral prism
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
icosahedral prism
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.png
icosidodecahedral prism
CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.png
Truncated dodecahedral prism
CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Tetrahedral prism.pngOctahedral prism.pngCuboctahedral prism.pngIcosahedral prism.pngIcosidodecahedral prism.pngTruncated dodecahedral prism.png
Rhombi-cosidodecahedral prism
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Rhombi-cuboctahedral prism
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Truncated cubic prism
CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Snub dodecahedral prism
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.png
n-gonal antiprismatic prism
CDel node h.pngCDel n.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
Rhombicosidodecahedral prism.pngRhombicuboctahedral prism.pngTruncated cubic prism.pngSnub dodecahedral prism.pngSquare antiprismatic prism.png
Cantellated 5-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cantitruncated 5-cell
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Runcinated 5-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcitruncated 5-cell
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cantellated tesseract
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cantitruncated tesseract
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Runcinated tesseract
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcitruncated tesseract
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
4-simplex t02.svg4-simplex t012.svg4-simplex t03.svg4-simplex t013.svg4-cube t02.svg4-cube t012.svg4-cube t03.svg4-cube t013.svg
Cantellated 24-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Cantitruncated 24-cell
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Runcinated 24-cell
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcitruncated 24-cell
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Cantellated 120-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cantitruncated 120-cell
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Runcinated 120-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcitruncated 120-cell
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
24-cell t02 F4.svg24-cell t012 F4.svg24-cell t03 F4.svg24-cell t013 F4.svg120-cell t02 H3.png120-cell t012 H3.png120-cell t03 H3.png120-cell t013 H3.png

See also[edit]

External links[edit]