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.

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.

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 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:

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

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:

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: [6,2], (*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
V62	V122	V62	V4.4.6	V26	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 mutations of omnitruncated tilings: 4.6.2n
Sym. *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
	*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

See also

Family of uniform prisms

Polyhedron
Coxeter
Tiling
Config.	3.4.4	4.4.4	5.4.4	6.4.4	7.4.4	8.4.4	9.4.4	10.4.4	11.4.4	12.4.4

External links

• Uniform Honeycombs in 3-Space VRML models
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra Prisms and antiprisms
• Weisstein, Eric W., "Hexagonal prism", MathWorld.
• Hexagonal Prism Interactive Model -- works in your web browser