# Pentagonal rotunda

Pentagonal rotunda
Type Johnson
J5 - J6 - J7
Faces 10 triangles
1+5 pentagons
1 decagon
Edges 35
Vertices 20
Vertex configuration 2.5(3.5.3.5)
10(3.5.10)
Symmetry group C5v
Rotation group C5, [5]+, (55)
Dual polyhedron -
Properties convex
Net

In geometry, the pentagonal rotunda is one of the Johnson solids (J6). It can be seen as half an icosidodecahedron.

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, surface area, circumradius and height can be used if all faces are regular, with edge length a:[2]

${\displaystyle V=\left({\frac {1}{12}}\left(45+17{\sqrt {5}}\right)\right)a^{3}\approx 6.91776...a^{3}}$

${\displaystyle A=\left({\frac {1}{2}}\left(5{\sqrt {3}}+{\sqrt {10\left(65+29{\sqrt {5}}\right)}}\right)\right)a^{2}=\left({\frac {1}{2}}{\sqrt {5\left(145+58{\sqrt {5}}+2{\sqrt {30\left(65+29{\sqrt {5}}\right)}}\right)}}\right)a^{2}\approx 22.3472...a^{2}}$

${\displaystyle R=\left({\frac {1}{2}}\left(1+{\sqrt {5}}\right)\right)a\approx 1.61803...a}$

${\displaystyle H=\left({\sqrt {1+{\frac {2}{\sqrt {5}}}}}\right)a\approx 1.37638...a}$

## Dual polyhedron

The dual of the pentagonal rotunda has 20 faces: 10 triangular, 5 rhombic, and 5 kites.

Dual pentagonal rotunda Net of dual

## 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 Rotunda" from Wolfram Alpha. Retrieved July 21, 2010.