Pentagonal orthocupolarotunda

In geometry, the pentagonal orthocupolarotunda is one of the Johnson solids ($J31 – J32 – J33$). As the name suggests, it can be constructed by joining a pentagonal cupola ($C5v$) and a pentagonal rotunda ($10(3.4.3.5) 5(3.4.5.4) 2.5(3.5.3.5)$) along their decagonal bases, matching the pentagonal faces. A 36-degree rotation of one of the halves before the joining yields a pentagonal gyrocupolarotunda ($J32$).

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


 * $$V=\frac{5}{12}\left(11+5\sqrt{5}\right)a^3\approx9.24181...a^3$$


 * $$A=\left(5+\frac{1}{4}\sqrt{1900+490\sqrt{5}+210\sqrt{75+30\sqrt{5}}}\right)a^2\approx23.5385...a^2$$