# 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}× |
| Wythoff symbol | 2 6 | 2 2 2 3 | |
| Coxeter diagrams | |
| Symmetry | D$_{6h}$, [6,2], (*622), order 24 |
| Rotation group | D$_6$, [6,2]$^+$, (622), order 12 |
| References | U$_{76(d)}$ |
| Dual | Hexagonal dipyramid |
| Properties | convex, zonohedron |
| Vertex figure 4.4.6 ||
In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.
Since it has 8 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 tilarity 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.

## 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 hexagonal prism is D$_{6h}$ of order 24. The rotation group is D$_6$ 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:
$V={\frac {3{\sqrt {3}}}{2}}a^{2}\times h$ and its surface area can be $S=3a({\sqrt {3}}a+2h)$.

## Symmetry

The topology of a uniform hexagonal prism can have geometric variations of lower symmetry, including:
| Name | Regular-hexagonal prism | Hexagonal frustum | Ditrigonal prism | Triambic prism | Ditrigonal trapezoprism |
| Symmetry | D$_{6h}$, [2,6], (*622) | C$_{6v}$, [6], (*66) | D$_{3h}$, [2,3], (*322) | D$_{3h}$, [2,3], (*322) | D$_{3d}$, [2$^+$,6], (2*3) |
| Construction | {6}×, | | t{3}×, | | s$_2${2,6}, |
| Image | | | | | |
| Distortion | | | | | |

## As part of spatial tesselations

It exists as cells of four prismatic uniform convex 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 4-polytopes, 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 |

## Related polyhedra and tilings

Uniform hexagonal dihedral spherical polyhedra :
| Symmetry:, (*622) | Symmetry:, (*622) | Symmetry:, (*622) | Symmetry:, (*622) | Symmetry:, (*622) | Symmetry:, (*622) | Symmetry:, (*622) | [6,2]$^+$, (622) | [6,2$^+$], (2*3) |
| {6,2} | t{6,2} | r{6,2} | t{2,6} | {2,6} | rr{6,2} | tr{6,2} | sr{6,2} | s{2,6} |
| Duals to uniforms ||||||||||
| V6$^2$ | V12$^2$ | V6$^2$ | V4.4.6 | V2$^6$ | V4.4.6 | V4.4.12 | V3.3.3.6 | 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.
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n vte :
| Sym. *n32 [n,3] | Spherical | Spherical | Spherical | Spherical | Euclid. | Compact hyperb. | Compact hyperb. | Paraco. | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic | Noncompact hyperbolic |
| Sym. *n32 [n,3] | *232 [2,3] | *332 [3,3] | *432 [4,3] | *532 [5,3] | *632 [6,3] | *732 [7,3] | *832 [8,3] | *∞32 [∞,3] | [12i,3] | [9i,3] | [6i,3] | [3i,3] |
| Figures | | | | | | | | | | | | |
| Config. | 4.6.4 | 4.6.6 | 4.6.8 | 4.6.10 | 4.6.12 | 4.6.14 | 4.6.16 | 4.6.∞ | 4.6.24i | 4.6.18i | 4.6.12i | 4.6.6i |
| Duals | | | | | | | | | | | | |
| Config. | V4.6.4 | V4.6.6 | V4.6.8 | V4.6.10 | V4.6.12 | V4.6.14 | V4.6.16 | V4.6.∞ | V4.6.24i | V4.6.18i | V4.6.12i | V4.6.6i |