# Hexagonal prism

Hexagonal prism
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymHip
Coxeter diagramx x6o ()
Stewart notationP6
Elements
Faces6 squares, 2 hexagons
Edges6+12
Vertices12
Vertex figureIsosceles triangle, edge lengths 3, 2, 2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt5}{2} ≈ 1.11803}$
Volume${\displaystyle \frac{3\sqrt3}{2} ≈ 2.59808}$
Dihedral angles4–4: 120°
4–6: 90°
Height1
Central density1
Number of pieces8
Level of complexity3
Related polytopes
ArmyHip
RegimentHip
DualHexagonal tegum
ConjugateNone
Abstract properties
Flag count72
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryG2×A1, order 24
ConvexYes
NatureTame

The hexagonal prism or hip is a prismatic uniform polyhedron. It consists of 2 hexagons and 6 squares. Each vertex joins one hexagon and two squares. As the name suggests, it is a prism based on a hexagon.

It can tile 3D space to form the hexagonal prismatic honeycomb.

## Vertex coordinates

A hexagonal prism of edge length 1 has vertex coordinates given by:

• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±1,\,0,\,±\frac12\right).}$

## Representations

A hexagonal prism has the following Coxeter diagrams:

## Semi-uniform variant

The hexagonal prism has a semi-uniform variant of the form x y6o that maintains its full symmetry. This variant uses rectangles as its sides.

With base edges of length a and side edges of length b, its circumradius is given by ${\displaystyle \sqrt{a^2+\frac{b^2}{4}}}$ and its volume is given by ${\displaystyle \frac{3\sqrt3}{2}a^2b}$.

A hexagonal prism with base edges of length a and side edges of length b can be atlernated to form a triangular antiprism with base edges of length ${\displaystyle \sqrt3a}$ and side edges of lengths ${\displaystyle \sqrt{a^2+b^2}}$. In particular if the side edges are ${\displaystyle \sqrt2}$ times the length of the base edges this gives a regular octahedron.

## Variations

A hexagonal prism has the following variations:

## Related polyhedra

A triangular cupola can be attached to a base of the hexagonal prism to form the elongated triangular cupola. If a second triangular cupola is attached to the other base in the same orientation, the result is the elongated triangular orthobicupola. If the second cupola is rotated 60º from the first cupola the result is the elongated triangular gyrobicupola.

It is also possible to augment square faces of the hexagonal prism with square pyramids. If one square is augmented the result is the augmented hexagonal prism. If a second square, opposite to the first,, is augmented the result is the parabiaugmented hexagonal prism. If two non-opposite, non-adjacent squares are augmented the result is the metabiaugmented hexagonal prism. If three mutually non-adjacent squares are augmented the result is the triaugmented hexagonal prism.

Two non-prismatic uniform polyhedron compounds are composed of hexagonal prisms:

There are also an infinite amount of prismatic uniform compounds that are the prisms of compounds of hexagons.