# Augmented hexagonal prism

Augmented hexagonal prism
Rank3
TypeCRF
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${\displaystyle {\frac {{\sqrt {2}}+9{\sqrt {3}}}{6}}\approx 2.83378}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\sqrt {\frac {7+2{\sqrt {6}}}{12}}}\right)\approx 174.73561^{\circ }}$
3–6: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
4–4: 120°
3–3: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
4–6: 90°
Central density1
Number of external pieces11
Level of complexity22
Related polytopes
ArmyAuhip
RegimentAuhip
DualLateromonotruncated hexagonal tegum
ConjugateAugmented hexagonal prism
Abstract & topological properties
Flag count88
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryK2×I, order 4
ConvexYes
NatureTame

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:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm 1,\,0,\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(0,\,{\frac {{\sqrt {2}}+{\sqrt {3}}}{2}},\,0\right).}$