# Pentagonal gyrocupolarotunda

Pentagonal gyrocupolarotunda
Type Johnson
J32 - J33 - J34
Faces 3×5 triangles
5 squares
2+5 pentagons
Edges 50
Vertices 25
Vertex configuration 10(32.4.5)
5(3.4.5.4)
2.5(3.5.3.5)
Symmetry group C5v
Dual polyhedron -
Properties convex
Net

In geometry, the pentagonal gyrocupolarotunda is one of the Johnson solids (J33). Like the pentagonal orthocupolarotunda (J32), it can be constructed by joining a pentagonal cupola (J5) and a pentagonal rotunda (J6) along their decagonal 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]

## Formulae

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

${\displaystyle V={\frac {5}{12}}(11+5{\sqrt {5}})a^{3}\approx 9.24181...a^{3}}$

${\displaystyle A=(5+{\frac {15}{4}}{\sqrt {3}}+{\frac {7}{4}}{\sqrt {25+10{\sqrt {5}}}})a^{2}\approx 23.5385...a^{2}}$

## 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 gyrocupolarotunda" from Wolfram Alpha. Retrieved July 24, 2010.