Elongated pentagonal cupola

In geometry, the elongated pentagonal cupola is one of the Johnson solids ($J19 – J20 – J21$). As the name suggests, it can be constructed by elongating a pentagonal cupola ($C5v$) by attaching a decagonal prism to its base. The solid can also be seen as an elongated pentagonal orthobicupola ($10(42.10) 10(3.43) 5(3.4.5.4)$) with its "lid" (another pentagonal cupola) removed.

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


 * $$V=\left(\frac{1}{6}\left(5+4\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)\right)a^3\approx10.0183...a^3$$


 * $$A=\left(\frac{1}{4}\left(60+\sqrt{10\left(80+31\sqrt{5}+\sqrt{2175+930\sqrt{5}}\right)}\right)\right)a^2\approx26.5797...a^2$$

Dual polyhedron
The dual of the elongated pentagonal cupola has 25 faces: 10 isosceles triangles, 5 kites, and 10 quadrilaterals.