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Concept
Triangular prism
Summary
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.
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, 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.
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 length of one side of the triangle, h is the length of an altitude drawn to that side, and l is the distance between the triangular faces.
A truncated right triangular prism has one triangular face truncated (planed) 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
There are two full D3h 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.
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