# Augmented pentagonal prism

Augmented pentagonal prism
Rank3
TypeCRF
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+4+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${\displaystyle {\frac {2{\sqrt {2}}+3{\sqrt {25+10{\sqrt {5}}}}}{12}}\approx 1.95618}$
Dihedral angles3–4 join: ${\displaystyle \arccos \left(-{\sqrt {\frac {13+{\sqrt {5}}+4{\sqrt {5-{\sqrt {5}}}}}{24}}}\right)\approx 162.73561^{\circ }}$
3–5 join: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
3–3 pyramidal: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
4–4: 108°
4–5: 90°
Central density1
Number of external pieces10
Level of complexity19
Related polytopes
ArmyAupip
RegimentAupip
DualLateromonotruncated pentagonal tegum
ConjugatesAugmented pentagrammic prism, Excavated pentagonal prism, Excavated pentagrammic prism
Abstract & topological properties
Flag count76
Euler characteristic2
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:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(0,\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(0,\,-{\frac {{\sqrt {2}}+{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}}{2}},\,0\right).}$