Triangular prism

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For the optical prism, see Triangular prism (optics).
Uniform Triangular prism
Triangular prism.png
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 diagram CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node.png
Symmetry group D3h, [3,2], (*322), order 12
Rotation group D3, [3,2]+, (322), order 6
References U76(a)
Dual Triangular dipyramid
Properties convex
Triangular prism vertfig.svg
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. 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[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:

where b is the triangle base length, h is the triangle height, and l is the length between the triangles.

Truncated triangular prism[edit]

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

TruncatedTriangularPrism.png

Facetings[edit]

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
Triangular prism.png FacetedTriangularPrism2.png FacetedTriangularPrism.png FacetedTriangularPrism3.png FacetedTriangularPrism4.png
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[edit]

Family of uniform prisms
Polyhedron Triangular prism.png Tetragonal prism.png Pentagonal prism.png Hexagonal prism.png Prism 7.png Octagonal prism.png Prism 9.png Decagonal prism.png Hendecagonal prism.png Dodecagonal prism.png
Coxeter CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 9.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 10.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 11.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 12.pngCDel node.pngCDel 2.pngCDel node 1.png
Tiling Spherical triangular prism.png Spherical square prism.png Spherical pentagonal prism.png Spherical hexagonal prism.png Spherical heptagonal prism.png Spherical octagonal prism.png Spherical decagonal prism.png
Config. 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
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}
Cupola 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)
Related
uniform
polyhedra
Triangular prism
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png
Cubocta-
hedron

CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Rhombi-
cubocta-
hedron

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Rhomb-
icosidodeca-
hedron

CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
Rhombi-
trihexagonal
tiling

CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png

Symmetry mutations[edit]

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.

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.

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.

Four dimensional space[edit]

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

See also[edit]

References[edit]

  1. ^ William F. Kern, James R Bland,Solid Mensuration with proofs, 1938, p.81