Gyroelongated pentagonal cupola

In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (J24). As the name suggests, it can be constructed by gyroelongating a pentagonal cupola (J5) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal bicupola (J46) with one pentagonal cupola removed.

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


 * $$A=\frac{1}{4}\left( 20+25\sqrt{3}+\left(10+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}\right)a^2\approx25.240003791...a^2,$$

and the volume is


 * $$V=\left(\frac{5}{6}+\frac{2}{3}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3\approx 9.073333194...a^3.$$

Dual polyhedron
The dual of the gyroelongated pentagonal cupola has 25 faces: 10 kites, 5 rhombi, and 10 pentagons.