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.

As a semiregular 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 prismatic symmetry $D_{6\mathrm{h}}$ of order 24, consisting of rotation around an axis passing through the regular hexagon bases' center, and reflection across a horizontal plane.

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: $$V=\frac{3\sqrt{3}}{2}{a}^{2}h,$$ and its surface area is by summing the area of two regular hexagonal bases and the lateral faces of six squares: $$S=3a(\sqrt{3}a+2h).$$

As a parallelohedron

File:Hexagonal prismatic honeycomb.png

Hexagonal prismatic honeycomb

The hexagonal prism is one of the parallelohedron, a polyhedral class that can be translated without rotations in Euclidean space, producing honeycombs; this class was discovered by Evgraf Fedorov in accordance with his studies of crystallography systems. The hexagonal prism is generated from four line segments, three of them parallel to a common plane and the fourth not. Its most symmetric form is the right prism over a regular hexagon, forming the hexagonal prismatic honeycomb.

The hexagonal prism also exists as cells of four prismatic uniform convex honeycombs in 3 dimensions:

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	File:Snub triangular-hexagonal prismatic honeycomb.png	File:Rhombitriangular-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 tetrahedral prism.png	File:Truncated octahedral prism.png	File:Truncated cuboctahedral prism.png	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 t013.svg	File:4-simplex t0123.svg	File:4-cube t023.svg	File:4-cube t0123.svg
runcitruncated 24-cell Template:CDD	omnitruncated 24-cell Template:CDD	runcitruncated 600-cell Template:CDD	omnitruncated 120-cell Template:CDD
File:24-cell t0123 F4.svg	File:24-cell t013 F4.svg	File:120-cell t023 H3.png	File:120-cell t0123 H3.png

External links

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