In geometry, a triangularprism 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. These three faces are parallelograms. All cross-sections parallel to the base faces are the same triangle.
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 tocompute the area of the triangle and multiply this by the length of the prism: where is the length of one side of the triangle, is the length of an altitude drawn to that side, and is the distance between the triangular faces.
Truncated triangular prism
A truncated right triangular prism has one triangular face truncated at an oblique angle. The volume of a truncated triangular prism with base area A and the three heights h1, h2, and h3 is determined by
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 lateralcrossed square faces, and 3 isosceles triangle lateral faces.
Related polyhedra and tilings
Symmetry mutations
This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations, and Coxeter group symmetry. This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure, and continues as tilings of the hyperbolic plane. These vertex-transitive figures have reflectional symmetry. This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure, and continues as tilings of the hyperbolic plane. These vertex-transitive figures have reflectional symmetry.
Compounds
There are 4 uniform compounds of triangular prisms:
Honeycombs
There are 9 uniform honeycombs that include triangular prism cells:
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. In Coxeter's notation the triangular prism is given the symbol −121.