# Biaugmented pentagonal prism

Biaugmented pentagonal prism Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymBaupip
Elements
Faces2+2+4 triangles, 1+2 squares, 2 pentagons
Edges1+2+2+2+4+4+4+4
Vertices2+2+2+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$\frac{4\sqrt2+3\sqrt{25+10\sqrt5}}{12} ≈ 2.19188$ 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
ArmyBaupip
RegimentBaupip
DualParalaterobitruncated pentagonal tegum
ConjugateBiaugmented pentagrammic prism
Abstract properties
Euler characteristic2
Topological properties
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:

• $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12\right),$ • $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac12\right),$ • $\left(0,\,±\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12\right),$ • $\left(±\frac{3+\sqrt5+2\sqrt{5+\sqrt5}}{8},\,-\sqrt{\frac{35-9\sqrt5+4\sqrt{25-5\sqrt5}}{160}},\,0\right).$ 