Pentagonal rotunda

In geometry, the pentagonal rotunda is one of the Johnson solids ($J5 – J6 – J7$). It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces.

Formulae
The following formulae for volume, surface area, circumradius, and height are valid if all faces are regular, with edge length a:
 * $$V=\left(\frac{1}{12}\left(45+17\sqrt{5}\right)\right)a^3\approx6.91776...a^3$$
 * $$\begin{align}

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\approx22.3472...a^2 \end{align}$$
 * $$R=\left(\frac{1}{2}\left(1+\sqrt{5}\right)\right)a\approx1.61803...a$$
 * $$H=\left(\sqrt{1+\frac{2}{\sqrt{5}}}\right)a\approx1.37638...a$$

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