# Biaugmented pentagonal prism

Biaugmented pentagonal prism
Rank3
TypeCRF
Notation
Bowers style acronymBaupip
Elements
Faces2+2+4 triangles, 1+2 squares, 2 pentagons
Edges1+2+2+2+4+4+4+4
Vertices2+2+4+4
Vertex figures2 square, edge length 1
4+4 irregular tetragons, edge lengths 1, 1, 2, (1+5)/2
2 isosceles triangles, edge lengths (1+5)/2, 2, 2
Measures (edge length 1)
Volume${\displaystyle {\frac {4{\sqrt {2}}+3{\sqrt {25+10{\sqrt {5}}}}}{12}}\approx 2.19188}$
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: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)144.73561^{\circ }}$
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 pieces13
Level of complexity23
Related polytopes
ArmyBaupip
RegimentBaupip
DualParalaterobitruncated pentagonal tegum
ConjugateBiaugmented pentagrammic prism
Abstract & topological properties
Flag count92
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryK2×I, order 4
ConvexYes
NatureTame

The biaugmented pentagonal prism is one of the 92 Johnson solids (J53). It consists of 2+2+4 triangles, 1+2 squares, and 2 pentagons. It can be constructed by attaching square pyramids to two non-adjacent square faces of the pentagonal prism.

## Vertex coordinates

A biaugmented pentagonal prism of edge length 1 has the following vertices:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(0,\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}+2{\sqrt {5+{\sqrt {5}}}}}{8}},\,-{\sqrt {\frac {35-9{\sqrt {5}}+4{\sqrt {25-5{\sqrt {5}}}}}{160}}},\,0\right).}$