Triangular prism

Triangular prism
Triangular prism
Type	Prism; Semiregular polyhedron; Uniform polyhedron
Faces	2 triangles; 3 squares
Edges	9
Vertices	6
Symmetry group	D3h
Dual polyhedron	Triangular bipyramid

In geometry, a triangular prism or trigonal prism^[1] is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform.

The triangular prism can be used in constructing another polyhedron. Examples are some of the Johnson solids, the truncated right triangular prism, and Schönhardt polyhedron.

Properties

A triangular prism has 6 vertices, 9 edges, and 5 faces. Every prism has 2 congruent faces known as its bases, and the bases of a triangular prism are triangles. The triangle has 3 vertices, each of which pairs with another triangle's vertex, making up another 3 edges. These edges form 3 parallelograms as other faces.^[2] If the prism's edges are perpendicular to the base, the lateral faces are rectangles, and the prism is called a right triangular prism.^[3] This prism may also be considered a special case of a wedge.^[4]

If the base is

internal angle of an equilateral triangle

π /3 = 60°

, and that between a square and a triangle is

π /2 = 90°

.^[7]

The volume of any prism is the product of the area of the base and the distance between the two bases.[8] In the case of a triangular prism, its base is a triangle, so its volume can be calculated by multiplying the area of a triangle and the length of the prism:

{\frac {bhl}{2}},

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.^[9] In the case of a right triangular prism, where all its edges are equal in length

l

, its volume can be calculated as the product of the equilateral triangle's area and length

l

:^[10]

{\frac {\sqrt {3}}{2}}l^{2}\cdot l\approx 0.433l^{3}

The triangular prism can be represented as the prism graph $Π 3$ . More generally, the prism graph $Π n$ represents the $n$ -sided prism.^[11]

Related polyhedron

In construction of polyhedron

Beyond the triangular bipyramid as its dual polyhedron, many other polyhedrons are related to the triangular prism. A

equilateral square pyramids onto the square face of the prism. The gyrobifastigium is constructed by attaching two triangular prisms along one of its square faces.^[13]

A truncated triangular prism is a triangular prism constructed by truncating its part at an oblique angle. As a result, the two bases are not parallel and every height has a different edge length. If the edges connecting bases are perpendicular to one of its bases, the prism is called a truncated right triangular prism. Given that $A$ is the area of the triangular prism's base, and the three heights $h 1$ , $h 2$ , and $h 3$ , its volume can be determined in the following formula:^[14]

{\frac {A(h_{1}+h_{2}+h_{3})}{3}}.

Schönhardt polyhedron is another polyhedron constructed from a triangular prism with equilateral triangle bases. This way, one of its bases rotates around the prism's centerline and breaks the square faces into skew polygons. Each square face can be re-triangulated with two triangles to form a non-convex dihedral angle.^[15] As a result, the Schönhardt polyhedron cannot be triangulated by a partition into tetrahedra. It is also that the Schönhardt polyhedron has no internal diagonals.^[16] It is named after German mathematician Erich Schönhardt, who described it in 1928, although the related structure was exhibited by artist Karlis Johansons in 1921.^[17]

There are 4 uniform compounds of triangular prisms. They are compound of four triangular prisms, compound of eight triangular prisms, compound of ten triangular prisms, compound of twenty triangular prisms.^[18]

Honeycombs

There are 9 uniform honeycombs that include triangular prism cells:

elongated triangular prismatic honeycomb

Related polytopes

The triangular prism is first in a dimensional series of

Coxeter

's notation the triangular prism is given the symbol −1₂₁.

k₂₁ figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
E_n	3	4	5	6	7	8	9	10
Coxeter group	E₃=A₂A₁	E₄=A₄	E₅=D₅	E₆	E₇	E₈	E₉ = ${\tilde {E}}_{8}$ = E₈⁺	E₁₀ = ${\bar {T}}_{8}$ = E₈⁺⁺
Coxeter diagram
Symmetry	[3^−1,2,1]	[3^0,2,1]	[3^1,2,1]	[3^2,2,1]	[3^3,2,1]	[3^4,2,1]	[3^5,2,1]	[3^6,2,1]
Order	12	120	1,920	51,840	2,903,040	696,729,600	∞
Graph							-	-
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 4-polytopes, including:

Four dimensional polytopes with triangular prisms
Tetrahedral prism	Octahedral prism	Cuboctahedral prism	Icosahedral prism	Icosidodecahedral prism	Truncated dodecahedral prism

Rhomb-icosidodecahedral prism	Rhombi-cuboctahedral prism	Truncated cubic prism	Snub dodecahedral prism	n-gonal antiprismatic prism

Cantellated 5-cell	Cantitruncated 5-cell	Runcinated 5-cell	Runcitruncated 5-cell	Cantellated tesseract	Cantitruncated tesseract	Runcinated tesseract	Runcitruncated tesseract

Cantellated 24-cell	Cantitruncated 24-cell	Runcinated 24-cell	Runcitruncated 24-cell	Cantellated 120-cell	Cantitruncated 120-cell	Runcinated 120-cell	Runcitruncated 120-cell