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.

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

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

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:

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

• 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