Triangular prism
| Uniform Triangular prism | |
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| 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 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:
where b is the triangle base length, h is the triangle height, and l is the length between the triangles.
Related polyhedra and tilings
| Symmetry | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
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| [2n,2] [n,2] [2n,2+] [2n+,2] |
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| As spherical polyhedra | ||||||||||
| Image |  |
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| 2 | 3 | 4 | 5 | 6 |
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 Digonal cupola |
 Triangular cupola |
 Square cupola |
 Pentagonal cupola |
 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.
| Symmetry *n32 [n,3] |
Spherical | Euclidean | Hyperbolic... | |||||
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| *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 |
3.4.4 |
 3.6.6 |
 3.8.8 |
 3.10.10 |
 3.12.12 |
 3.14.14 |
 3.16.16 |
 3.∞.∞ |
| Coxeter Schläfli |
t{2,3} |
t{3,3} |
t{4,3} |
t{5,3} |
t{6,3} |
t{7,3} |
t{8,3} |
 t{∞,3} |
| Uniform dual figures | ||||||||
| Triakis figures |
 V3.4.4 |
 V3.6.6 |
 V3.8.8 |
 V3.10.10 |
 V3.12.12 |
 V3.14.14 |
 V3.16.16 |
 V3.∞.∞ |
| Coxeter |  |
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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.
| Symmetry *n32 [n,3] |
Spherical | Euclidean | Hyperbolic... | |||||
|---|---|---|---|---|---|---|---|---|
| *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 |
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 |
| Coxeter Schläfli |
rr{2,3} |
rr{3,3} |
rr{4,3} |
rr{5,3} |
rr{6,3} |
rr{7,3} |
rr{8,3} |
rr{∞,3} |
| Deltoidal figure | V3.4.2.4 |
 V3.4.3.4 |
 V3.4.4.4 |
 V3.4.5.4 |
 V3.4.6.4 |
 V3.4.7.4 |
 V3.4.8.4 |
 V3.4.∞.4 |
| Coxeter | |
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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.
| En | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
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| Coxeter group |
E3=A2×A1 | E4=A4 | E5=D5 | E6 | E7 | E8 | E9 =  = E8+ |
E10 = E8++ |
| Coxeter diagram |
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| 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] (∞) |
| Graph | |
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∞ | ∞ |
| 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 polychora, including:
| tetrahedral prism |
octahedral prism |
cuboctahedral prism |
icosahedral prism |
icosidodecahedral prism |
Truncated dodecahedral prism | ||
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| Rhombi-cosidodecahedral prism |
Rhombi-cuboctahedral prism |
Truncated cubic prism |
Snub dodecahedral prism |
n-gonal antiprismatic prism  | |||
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| Cantellated 5-cell |
Cantitruncated 5-cell |
Runcinated 5-cell |
Runcitruncated 5-cell |
Cantellated tesseract |
Cantitruncated tesseract |
Runcinated tesseract |
Runcitruncated tesseract |
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| Cantellated 24-cell |
Cantitruncated 24-cell |
Runcinated 24-cell |
Runcitruncated 24-cell |
Cantellated 120-cell |
Cantitruncated 120-cell |
Runcinated 120-cell |
Runcitruncated 120-cell |
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See also
External links
- Weisstein, Eric W., "Triangular prism", MathWorld.
- Interactive Polyhedron: Triangular Prism
- Whole site dedicated to triangular prisms. Good resource for high school students to learn about how to find the surface area and volume of a triangular prism and how to draw its net.