# Triangular prism
A triangular prism is a three-dimensional polyhedron with two parallel, congruent triangular faces (the bases) joined by three rectangular or [parallelogram](https://www.edgepedia.org/parallelogram) side faces, giving 5 faces, 9 edges, and 6 vertices.[1](https://bhanzu.com/math/geometry/triangular-prism) It is the prism family member whose base is a triangle, and in a right triangular prism the three lateral faces are rectangles perpendicular to the triangular bases; if they are not perpendicular, the prism is simply oblique.[2](https://www.cut-the-knot.org/Curriculum/Geometry/Polyhedra/3prism.shtml)
| Key fact | Detail |
|---|---|
| Counting | 5 faces (2 triangular bases, 3 lateral faces), 9 edges, 6 vertices; Euler's formula holds: 6 − 9 + 5 = 2[1](https://bhanzu.com/math/geometry/triangular-prism) • [2](https://www.cut-the-knot.org/Curriculum/Geometry/Polyhedra/3prism.shtml) |
| Volume | Base area times altitude; the triangular prism is half of a parallelepiped with the same base and altitude[3](https://www.gutenberg.org/files/21076/21076-h/21076-h.htm) |
| Surface area | For a right prism, SA = 2B + ph, where B is the base area, p the base perimeter, and h the height[4](https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area) |
| Dual polyhedron | The triangular dipyramid, two tetrahedra joined at a face[5](https://usawebschool.org/math/glossary/triangular-prism) |
| Symmetry | The right triangular prism with equilateral bases has symmetry group D3h of order 12[6](https://sivo.it.com/geometric-symmetry/what-is-the-symmetry-of-a-regular-triangular-prism/) |
| Graph skeleton | The triangular prism graph Y3 has 6 nodes and 9 edges and is the line graph of the complete bipartite graph K2,3[7](https://mathworld.wolfram.com/PrismGraph.html) |
| Tiling | A convex polyhedron is a universal tiler if and only if it is a tetrahedron or a triangular prism[8](https://arxiv.org/abs/1110.1715) |
## Geometry and counting
The two triangular faces are the bases, and the three faces joining them are the lateral faces; the edges of the lateral faces are called lateral edges.[2](https://www.cut-the-knot.org/Curriculum/Geometry/Polyhedra/3prism.shtml) When the triangular bases are equilateral, the prism is called regular, and its rectangular faces are congruent.[2](https://www.cut-the-knot.org/Curriculum/Geometry/Polyhedra/3prism.shtml) Like every convex polyhedron, the triangular prism satisfies [Euler's formula](https://www.edgepedia.org/eulers-formula), F + V − E = 2, which for this solid reads 5 + 6 − 9 = 2.[9](https://crazymathsgames.com/shapes/triangular-prism/)
The solid can be described algebraically as the [Cartesian product](https://www.edgepedia.org/cartesian-product) of a triangle with a line segment.[5](https://usawebschool.org/math/glossary/triangular-prism) This product view explains the counts directly: two copies of the triangle give the bases, and each of the triangle's 3 edges sweeps out one rectangular lateral face.
## Volume and surface area
The volume of a triangular prism is the product of its base area and its altitude. The classical derivation treats the prism as half of a parallelepiped having a double base and equal altitude, so a diagonal plane of a parallelepiped divides it into two prisms of equal volume.[3](https://www.gutenberg.org/files/21076/21076-h/21076-h.htm) In symbols, with A the area of the triangular base and l the prism's length,
\[ V = A_{\triangle} \cdot l. \]
The same base-times-height rule holds for right and oblique prisms alike, since any cross section parallel to the bases is congruent to them.[10](https://www.mathopenref.com/prism.html)
For surface area, the general right-prism formula is SA = 2B + ph, where B is the area of one base, p the perimeter of the base, and h the height.[4](https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area) For a triangular prism this means two triangular faces plus three rectangles whose total area is the triangle's perimeter times the prism length.[5](https://usawebschool.org/math/glossary/triangular-prism) A worked example from OpenStax: a right triangular prism with base legs 8.49 in and 8.49 in, base side 12 in, triangle height 6 in, and prism width 10 in has base area 36 in², perimeter 28.98 in, and surface area 2(36) + 28.98(10) = 361.8 in².[4](https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area)
## Definition in Euclid
Euclid's *Elements*, Book XI, defines a prism as "a solid figure contained by planes two of which, namely those which are opposite, are equal, similar, and parallel, while the rest are parallelograms," and gives EFGHKL as a prism with opposite triangular sides EFG and HKL.[11](https://aleph0.clarku.edu/~djoyce/java/elements/bookXI/defXI12.html) The most ancient available copy of the *Elements* is a 9th-century Greek manuscript in the Vatican library, some eleven centuries after Euclid's death.[12](https://link.springer.com/article/10.1007/s00004-024-00772-1) Later textbook geometry built directly on this foundation: the 19th-century Byrne-style edition states that the volume of a triangular prism equals the product of its base and altitude because it is half of a parallelepiped, and that any prism can be divided into triangular prisms.[3](https://www.gutenberg.org/files/21076/21076-h/21076-h.htm)
## The triangular prism graph
The skeleton of a triangular prism is the 3-prism graph Y3, one of the prism graphs, which correspond to the skeletons of n-prisms. An n-prism graph has 2n nodes and 3n edges, is planar and polyhedral, and is isomorphic to the generalized Petersen graph P(n,1) and to the Cartesian product P2 □ Cn.[7](https://mathworld.wolfram.com/PrismGraph.html) For n = 3 this gives 6 nodes and 9 edges, matching the solid's counts; Y3 is the line graph of K2,3, while Y4 is the cubical graph of the ordinary cube.[7](https://mathworld.wolfram.com/PrismGraph.html) Every prism graph is cubic, since each vertex carries two cycle edges and one cross-layer edge.
The triangular prism graph appears as an exceptional case in coloring theory. A proof concerning Kempe equivalence of colorings, the reconfiguration of colorings by exchanging colors along two-colored connected subgraphs, shows that all 3-colorings of a cubic graph are Kempe equivalent unless the graph is the complete graph K4 or the triangular prism; the 3-prism is the only counterexample for the case k = 3.[13](https://arxiv.org/abs/1503.03430) The triangular prism also belongs to the class of graphs with representation number 3, and in fact every prism belongs to that class.[14](https://arxiv.org/abs/1403.1616)
## Symmetry
A right triangular prism with equilateral triangular bases has the three-dimensional symmetry group D3h, of order 12.[6](https://sivo.it.com/geometric-symmetry/what-is-the-symmetry-of-a-regular-triangular-prism/) In graph terms, the prism graph is the [Cayley graph](https://www.edgepedia.org/cayley-graph) of the dihedral group with respect to a suitable generating set, and for the triangular prism the group is D3 = S3.[7](https://mathworld.wolfram.com/PrismGraph.html) • [15](https://www.jaapsch.net/puzzles/cayley.htm)
## Tiling and related solids
A 2011 result on universal tilers—convex polyhedra that tile space in a way that lets any specified polyhedron fit—establishes that the only convex polyhedra with this property are tetrahedra and triangular prisms.[8](https://arxiv.org/abs/1110.1715) Its dual, the triangular dipyramid, is formed by two tetrahedra joined at a face.[5](https://usawebschool.org/math/glossary/triangular-prism)
## Optical prisms
In optics, a triangular glass prism refracts light at its faces; a beam of white light passing through one breaks into its component wavelengths, producing the characteristic rainbow.[10](https://www.mathopenref.com/prism.html) [Total internal reflection](https://www.edgepedia.org/total-internal-reflection) can redirect light through multiple bounces inside a triangular prism geometry, a standard classroom demonstration with plexiglass blocks, where the critical angle for the plexiglass-air interface is about 42 degrees.[16](https://physicslabs.colorado.edu/demos/optics/geometrical-optics/total-internal-reflection/blackboard-optics-total-internal-reflection/) [Equilateral triangle](https://www.edgepedia.org/equilateral-triangle) prisms are used in spectral equipment, and the Dove prism, used in surveying instruments on sloping ground, is a trapezoidal shape formed by largely truncating the apex of a triangular prism.[17](https://www.nitto-optical.co.jp/english/products/basic-prism/) High-accuracy refractometry at NIST measures refractive index by the minimum-deviation-angle method using prisms of sizes from about 1 mm to 60 mm, with goniometer angle accuracy of 0.2 arc-sec.[18](https://www.nist.gov/laboratories/tools-instruments/minimum-deviation-angle-refractometry-system)
## References
1. [Triangular Prism - Volume, Surface Area, Formulas, Bhanzu](https://bhanzu.com/math/geometry/triangular-prism)
2. [Right Triangular Prism, Cut-the-Knot](https://www.cut-the-knot.org/Curriculum/Geometry/Polyhedra/3prism.shtml)
3. [First Six Books of the Elements of Euclid, Project Gutenberg](https://www.gutenberg.org/files/21076/21076-h/21076-h.htm)
4. [10.7 Volume and Surface Area, Contemporary Mathematics, OpenStax](https://openstax.org/books/contemporary-mathematics/pages/10-7-volume-and-surface-area)
5. [Triangular prism definition, USA Web School](https://usawebschool.org/math/glossary/triangular-prism)
6. [What is the Symmetry of a Regular Triangular Prism?, Sivo](https://sivo.it.com/geometric-symmetry/what-is-the-symmetry-of-a-regular-triangular-prism/)
7. [Prism Graph, Wolfram MathWorld](https://mathworld.wolfram.com/PrismGraph.html)
8. [Determining All Universal Tilers, arXiv](https://arxiv.org/abs/1110.1715)
9. [Triangular Prism: Definition, Net and Properties, CrazyMathsGames](https://crazymathsgames.com/shapes/triangular-prism/)
10. [Prism definition, Math Open Reference](https://www.mathopenref.com/prism.html)
11. [Euclid's Elements, Book XI, Definitions 12 and 13, Clark University](https://aleph0.clarku.edu/~djoyce/java/elements/bookXI/defXI12.html)
12. [The Regular Polyhedra: Drawing and Computing in Euclid's day, Nexus Network Journal](https://link.springer.com/article/10.1007/s00004-024-00772-1)
13. [Kempe Equivalence of Colourings of Cubic Graphs, arXiv](https://arxiv.org/abs/1503.03430)
14. [On graphs with representation number 3, arXiv](https://arxiv.org/abs/1403.1616)
15. [Cayley Graphs, Jaap's Puzzle Page](https://www.jaapsch.net/puzzles/cayley.htm)
16. [Blackboard Optics Total Internal Reflection, University of Colorado Boulder](https://physicslabs.colorado.edu/demos/optics/geometrical-optics/total-internal-reflection/blackboard-optics-total-internal-reflection/)
17. [Prisms product lineup, NITTO OPTICAL CO., LTD.](https://www.nitto-optical.co.jp/english/products/basic-prism/)
18. [Minimum-Deviation-Angle Refractometry System, NIST](https://www.nist.gov/laboratories/tools-instruments/minimum-deviation-angle-refractometry-system)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › General discrete mathematics and discrete structures › Combinatorics › Geometric, polyhedral, and topological combinatorics › Named polytopes and polytope families*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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