Hexagonal prism

Uniform Hexagonal prism
Type	Prismatic uniform polyhedron
Elements	F = 8, E = 18, V = 12 (χ = 2)
Faces by sides	6{4}+2{6}
Schläfli symbol	t{2,6} or {6}x{}
Wythoff symbol	2 6 | 2 2 2 3 |
Coxeter-Dynkin
Symmetry	D6h, [6,2], (*66)
References	U76(d)
Dual	Hexagonal dipyramid
Properties	convex, zonohedron
Vertex figure 4.4.6

In geometry, the hexagonal prism is a prism with hexagonal base. This polyhedron has 8 faces, 18 edges, and 12 vertices.

Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces. Because of the ambiguity of the term octahedron and the dissimilarity of the various eight-sided figures, the term is rarely used without clarification.

As a semiregular (or uniform) polyhedron

If faces are all regular, the hexagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the fourth in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a truncated hexagonal hosohedron, represented by Schläfli symbol t{2,6}. Alternately it can be seen as the Cartesian product of a regular hexagon and a line segment, and represented by the product {6}x{}. The dual of a hexagonal prism is a hexagonal bipyramid.

The symmetry group of a right pentagonal prism is D_6h of order 24. The rotation group is D₆ of order 12.

Volume

As in most prisms, the volume is found by taking the area of the base, with a side length of $a$, and multiplying it by the height $h$.

$V = \frac{3 \sqrt{3}}{2}a^2 \times h$

Other images

A translucent model

Use

It exists as cells of four prismatic uniform honeycombs in 3 dimensions:

Hexagonal prismatic honeycomb	Triangular-hexagonal prismatic honeycomb	Snub triangular-hexagonal prismatic honeycomb	Rhombitriangular-hexagonal prismatic honeycomb

It also exists as cells of a number of four-dimensional uniform polychora, including:

truncated tetrahedral prism	truncated octahedral prism	Truncated cuboctahedral prism	Truncated icosahedral prism	Truncated icosidodecahedral prism
runcitruncated 5-cell	omnitruncated 5-cell	runcitruncated 16-cell	omnitruncated tesseract
runcitruncated 24-cell	omnitruncated 24-cell	runcitruncated 600-cell	omnitruncated 120-cell

See also

• Set of prisms
• Triangular prism
• Cube Square-capped prism
• Pentagonal prism

External links

• Weisstein, Eric W., "Hexagonal prism" from MathWorld.
• Hexagonal Prism Interactive Model -- works in your web browser