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
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.

The 92 Johnson solids were named and described by Norman Johnson in 1966.

Formulae

The following formulae for volume, surface area, and circumradius can be used if all faces are regular, with edge length a:^[1]

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

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

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

Dual polyhedron

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

Dual pentagonal rotunda	Net of dual

Reference