# Augmented hexagonal prism

Augmented hexagonal prism
Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymAuhip
Coxeter diagramoxxx oxux&#xt
Elements
Faces2+2 triangles, 1+2+2 squares, 2 hexagons
Edges2+2+2+2+2+4+4+4
Vertices1+4+4+4
Vertex figures1 square, edge length 1
4 irregular tetragons, edge lengths 1, 1, 2, 3
4+4 isosceles triangles, edge lengths (1+5)/2, 2, 2
Measures (edge length 1)
Volume${\displaystyle \frac{\sqrt2+9\sqrt3}{6} ≈ 2.83378}$
Dihedral angles3–4: ${\displaystyle \arccos\left(-\sqrt{\frac{7+2\sqrt6}{12}}\right) ≈ 174.73561^\circ}$
3–6: ${\displaystyle \arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561^\circ}$
4–4: 120°
3–3: ${\displaystyle \arccos\left(-\frac13\right) ≈ 109.47122^\circ}$
4–6: 90°
Central density1
Related polytopes
ArmyAuhip
RegimentAuhip
DualLateromonotruncated hexagonal tegum
ConjugateAugmented hexagonal prism
Abstract properties
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryK2×I, order 4
ConvexYes
NatureTame
Discovered by{{{discoverer}}}

The augmented hexagonal prism is one of the 92 Johnson solids (J54). It consists of 2+2 triangles, 1+2+2 squares, and 2 hexagons. It can be constructed by attaching a square pyramid to one of the square faces of the hexagonal prism.

## Vertex coordinates

An augmented hexagonal prism of edge length 1 has the following vertices:

• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±1,\,0,\,±\frac12\right),}$
• ${\displaystyle \left(0,\,\frac{\sqrt2+\sqrt3}{2},\,0\right).}$