# 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 diagrams

Symmetry D6h, [6,2], (*622), order 24
Rotation group D6, [6,2]+, (622), order 12
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.[1]

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.

Before sharpening, many pencils take the shape of a long hexagonal prism.[2]

## 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}×{}. The dual of a hexagonal prism is a hexagonal bipyramid.

The symmetry group of a right pentagonal prism is D6h 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 $h$, giving the formula:[3]

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

## Symmetry

The topology of a uniform hexagonal prism can have geometric variations of lower symmetry, including:

Symmetry D6h, [2,6], (*622) C6v, [6], (*66) D3h, [2,3], (*322) D3d, [2+,6], (2*3)
Construction {6}×{}, t{3}×{}, s2{2,6},
Image
Distortion

## As part of spatial tesselations

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

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

## Related polyhedra and tilings

Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622) [6,2]+, (622) [1+,6,2], (322) [6,2+], (2*3)
{6,2} t{6,2} r{6,2} 2t{6,2}=t{2,6} 2r{6,2}={2,6} rr{6,2} tr{6,2} sr{6,2} h{6,2} s{2,6}
Uniform duals
V62 V122 V62 V4.4.6 V26 V4.4.6 V4.4.12 V3.3.3.6 V32 V3.3.3.3

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

Dimensional family of omnitruncated polyhedra and tilings: 4.6.2n
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Coxeter
Schläfli

tr{2,3}

tr{3,3}

tr{4,3}

tr{5,3}

tr{6,3}

tr{7,3}

tr{8,3}

tr{∞,3}
Omnitruncated
figure
Vertex figure 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞
Dual figures
Coxeter
Omnitruncated
duals
Face
configuration
V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞

Family of uniform prisms
Symmetry 3 4 5 6 7 8 9 10 11 12
[2n,2]
[n,2]
[2n,2+]

Image

As spherical polyhedra
Image

## References

1. ^ a b Pugh, Anthony (1976), Polyhedra: A Visual Approach, University of California Press, pp. 21, 27, 62, ISBN 9780520030565.
2. ^ Simpson, Audrey (2011), Core Mathematics for Cambridge IGCSE, Cambridge University Press, pp. 266–267, ISBN 9780521727921.
3. ^ Wheater, Carolyn C. (2007), Geometry, Career Press, pp. 236–237, ISBN 9781564149367.