Elongated pentagonal orthobirotunda

In geometry, the elongated pentagonal orthobirotunda is one of the Johnson solids ($J41 – J42 – J43$). Its Conway polyhedron notation is at5jP5. As the name suggests, it can be constructed by elongating a pentagonal orthobirotunda ($D5h$) by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae ($20(3.42.5) 2.10(3.5.3.5)$) through 36 degrees before inserting the prism yields the elongated pentagonal gyrobirotunda ($J42$).

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


 * $$V=\frac{1}{6}\left(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3\approx21.5297...a^3$$


 * $$A=\left(10+\sqrt{30\left(10+3\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2\approx39.306...a^2$$