Hexagonal prism

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.

As in most prisms, the volume is found by taking the area of the base, with a side length of ${\displaystyle a}$ , and multiplying it by the height ${\displaystyle h}$ , giving the formula:^[3]

${\displaystyle V={\frac {3{\sqrt {3}}}{2}}a^{2}\times h}$ and its surface area can be${\displaystyle S=3a({\sqrt {3}}a+2h)}$ .

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	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

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

Hexagonal prismatic honeycomb[1]	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

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 mutation of omnitruncated tilings: 4.6.2n v t e
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

• 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