Hexagonal prism

In geometry, the hexagonal prism is a prism with hexagonal base. this polyhedron has 8 faces, 18 edges, and 12 vertices.

Hexagon prism
Type	prism, parallelohedron
Symmetry group	prismatic symmetry ${\displaystyle D_{6\mathrm {h} }}$ of order 24
Dual polyhedron	hexagonal bipyramid

3D model of a uniform hexagonal prism.

Properties

A hexagonal prism has twelve vertices, eighteen edges, and eight faces. Every prism has two faces known as its bases, and the bases of a hexagonal prism are hexagons. The hexagons has six vertices, each of which pairs with another hexagon's vertex, forming six edges. These edges form three parallelograms as other faces.^[1] A prism is said to be right if the edges are of the same length and perpendicular to the base.

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 symmetry group of a right hexagonal prism is prismatic symmetry ${\displaystyle D_{6\mathrm {h} }}$ of order 24, consisting of rotation around an axis passing through the regular hexagon bases' center, and reflection across a horizontal plane.^[2] The dual of a hexagonal prism is a hexagonal bipyramid, both of which have the same three-dimensional symmetry group.

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

Honeycombs

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.^[4] Its most symmetric form is the right prism over a regular hexagon, forming the hexagonal prismatic honeycomb.^[5]

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

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