# Hexagonal antiprism

Hexagonal antiprism
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymHap
Coxeter diagrams2s12o
Elements
Faces12 triangles, 2 hexagons
Edges12+12
Vertices12
Vertex figureIsosceles trapezoid, edge lengths 1, 1, 1, 3
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{3+\sqrt3}}{2} ≈ 1.08766}$
Volume${\displaystyle \sqrt{2+2\sqrt3} ≈ 2.33754}$
Dihedral angles3–3: ${\displaystyle \arccos\left(\frac{1-2\sqrt3}{3}\right) ≈ 145.22189°}$
6–3: ${\displaystyle \arccos\left(\frac{3-2\sqrt3}{3}\right) ≈ 98.89943°}$
Height${\displaystyle \sqrt{\sqrt3-1} ≈ 0.85560}$
Central density1
Related polytopes
ArmyHap
RegimentHap
DualHexagonal antitegum
ConjugateHexagonal retroprism
Abstract properties
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(12)×A1)/2, order 24
ConvexYes
NatureTame

The hexagonal antiprism, or hap, is a prismatic uniform polyhedron. It consists of 12 triangles and 2 hexagons. Each vertex joins one hexagon and three triangles. As the name suggests, it is an antiprism based on a hexagon.

## Vertex coordinates

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

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

## Representations

A hexagonal antiprism has the following Coxeter diagrams:

## General variant

The hexagonal antiprism has a general isogonal variant of the form xo6ox&#y that maintains its full symmetry. This variant uses isosceles triangles as sides.

If the base edges are of length b and the lacing edges are of length l, its height is given by ${\displaystyle \sqrt{l^2-b^2(2-\sqrt3)}}$.

The bases of the pentagonal antiprism are rotated from each other by an angle of 30°. If this angle is changed the result is more properly called a hexagonal gyroprism.

A notable case occurs as the alternation of the uniform dodecagonal prism. This specific case has base edges of length ${\displaystyle \sqrt{2+\sqrt3}}$ and side edges of length ${\displaystyle \sqrt2}$.

## Related polyhedra

A triangular cupola can be attached to a base of the hexagonal antiprism to form the gyroelongated triangular cupola. If a second triangular cupola is attached to the other base, the result is the gyroelongated triangular bicupola.