Gyroelongated pentagonal rotunda

In geometry, the gyroelongated pentagonal rotunda is one of the Johnson solids (J25). As the name suggests, it can be constructed by gyroelongating a pentagonal rotunda (J6) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal birotunda (J48) with one pentagonal rotunda removed.

Area and Volume
With edge length a, the surface area is


 * $$A=\frac{1}{2}\left( 15\sqrt{3}+\left(5+3\sqrt{5}\right)\sqrt{5+2\sqrt{5}}\right)a^2\approx31.007454303...a^2,$$

and the volume is


 * $$V=\left(\frac{45}{12}+\frac{17}{12}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3\approx13.667050844...a^3.$$

Dual polyhedron
The dual of the gyroelongated pentagonal rotunda has 30 faces: 10 pentagons, 10 rhombi, and 10 quadrilaterals.