Hexagonal prism

File:Prisma hexagonal 3D.stl

3D model of a uniform hexagonal prism.

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 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:

$V=\frac{3\sqrt{3}}{2}{a}^2\times h$ 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	D6h, [2,6], (*622)	C6v, [6], (*66)	D3h, [2,3], (*322)		D3d, [2+,6], (2*3)
Construction	{6}×{}, Template:CDD		t{3}×{}, Template:CDD	Template:CDD	s2{2,6}, Template:CDD
Image	File:Hexagonal Prism.svg		File:Truncated triangle prism.png
Distortion				Error creating thumbnail:	File:Cantic snub hexagonal hosohedron2.png

As part of spatial tesselations

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

Hexagonal prismatic honeycomb Template:CDD	Triangular-hexagonal prismatic honeycomb Template:CDD	Snub triangular-hexagonal prismatic honeycomb Template:CDD	Rhombitriangular-hexagonal prismatic honeycomb Template:CDD
	File:Triangular-hexagonal prismatic honeycomb.png

It also exists as cells of a number of four-dimensional uniform 4-polytopes, including:

truncated tetrahedral prism Template:CDD	truncated octahedral prism Template:CDD	Truncated cuboctahedral prism Template:CDD	Truncated icosahedral prism Template:CDD	Truncated icosidodecahedral prism Template:CDD
			File:Truncated icosahedral prism.png	File:Truncated icosidodecahedral prism.png
runcitruncated 5-cell Template:CDD	omnitruncated 5-cell Template:CDD	runcitruncated 16-cell Template:CDD	omnitruncated tesseract Template:CDD
	File:4-simplex t0123.svg	Error creating thumbnail:	Error creating thumbnail:
runcitruncated 24-cell Template:CDD	omnitruncated 24-cell Template:CDD	runcitruncated 600-cell Template:CDD	omnitruncated 120-cell Template:CDD
	Error creating thumbnail:	File:120-cell t023 H3.png	Error creating thumbnail:

Related polyhedra and tilings

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram Template:CDD. 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.

External links

• Uniform Honeycombs in 3-Space VRML models
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra Prisms and antiprisms
• Template:Mathworld
• Hexagonal Prism Interactive Model -- works in your web browser