# Pentagonal gyrobicupola

Pentagonal gyrobicupola
Type Johnson
J30 - J31 - J32
Faces 10 triangles
10 squares
2 pentagons
Edges 40
Vertices 20
Vertex configuration 10(3.4.3.4)
10(3.4.5.4)
Symmetry group D5d
Dual polyhedron Polyhedron with faces in plane of the rhombic icosahedron
Properties convex
Net

In geometry, the pentagonal gyrobicupola is one of the Johnson solids (J31). Like the pentagonal orthobicupola (J30), it can be obtained by joining two pentagonal cupolae (J5) along their bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another.

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]

The pentagonal gyrobicupola is the third in an infinite set of gyrobicupolae.

The pentagonal gyrobicupola is what you get when you take a rhombicosidodecahedron, chop out the middle parabidiminished rhombicosidodecahedron (J80), and paste the two opposing cupolae back together.

## Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a:[2]

${\displaystyle V={\frac {1}{3}}(5+4{\sqrt {5}})a^{3}\approx 4.64809...a^{3}}$

${\displaystyle A=(10+{\sqrt {{\frac {5}{2}}(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}})}})a^{2}\approx 17.7711...a^{2}}$

## References

1. ^ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. ^ Stephen Wolfram, "Pentagonal gyrobicupola" from Wolfram Alpha. Retrieved July 24, 2010.