# Augmented pentagonal prism

Augmented pentagonal prism Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymAupip
Coxeter diagramoxxx oxfo&#xt
Elements
Faces2+2 triangles, 2+2 squares, 2 pentagons
Edges1+2+2+2+4+4+4
Vertices1+2+2+4
Vertex figures1 square, edge length 1
4 irregular tetragons, edge lengths 1, 1, 2, (1+5)/2
2+4 isosceles triangles, edge lengths (1+5)/2, 2, 2
Measures (edge length 1)
Volume$\frac{2\sqrt2+3\sqrt{25+10\sqrt5}}{12} ≈ 1.95618$ Dihedral angles3–4 join: $\arccos\left(-\sqrt{\frac{13+\sqrt5+4\sqrt{5-\sqrt5}}{24}}\right) ≈ 162.73561°$ 3–5 join: $\arccos\left(-\frac{\sqrt6}{3}\right) 144.73561°$ 3–3 pyramidal: $\arccos\left(-\frac13\right) ≈ 109.47122°$ 4–4: 108°
4–5: 90°
Central density1
Related polytopes
ArmyAupip
RegimentAupip
DualLateromonotruncated pentagonal tegum
ConjugateAugmented pentagrammic prism
Abstract properties
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryK2×I, order 4
ConvexYes
NatureTame

The augmented pentagonal prism is one of the 92 Johnson solids (J52). It consists of 2+2 triangles, 2 +2squares, and 2 pentagons. It can be constructed by attaching a square pyramid to one of the square faces of the pentagonal prism.

## Vertex coordinates

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

• $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12\right),$ • $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac12\right),$ • $\left(0,\,±\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12\right),$ • $\left(0,\,-\frac{\sqrt2+\sqrt{\frac{5+2\sqrt5}{5}}}{2},\,0\right).$ 