# Pentagonal rotunda

Pentagonal rotunda TypeJohnson
J5 - J6 - J7
Faces10 triangles
1+5 pentagons
1 decagon
Edges35
Vertices20
Vertex configuration2.5(3.5.3.5)
10(3.5.10)
Symmetry groupC5v
Rotation groupC5, +, (55)
Dual polyhedron-
Propertiesconvex
Net In geometry, the pentagonal rotunda is one of the Johnson solids (J6). It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda.

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.

## Formulae

The following formulae for volume, surface area, circumradius, and height are valid if all faces are regular, with edge length a:

$V=\left({\frac {1}{12}}\left(45+17{\sqrt {5}}\right)\right)a^{3}\approx 6.91776...a^{3}$ {\begin{aligned}A&=\left({\frac {1}{2}}{\sqrt {5\left(145+58{\sqrt {5}}+2{\sqrt {30\left(65+29{\sqrt {5}}\right)}}\right)}}\right)a^{2}\\&=\left({\frac {1}{2}}\left(5{\sqrt {3}}+{\sqrt {10\left(65+29{\sqrt {5}}\right)}}\right)\right)a^{2}\approx 22.3472...a^{2}\end{aligned}} $R=\left({\frac {1}{2}}\left(1+{\sqrt {5}}\right)\right)a\approx 1.61803...a$ $H=\left({\sqrt {1+{\frac {2}{\sqrt {5}}}}}\right)a\approx 1.37638...a$ ## Dual polyhedron

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