Gyroelongated pentagonal birotunda

In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids ($J47 – J48 – J49$). As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda (either $D5$ or the icosidodecahedron) by inserting a decagonal antiprism between its two halves.

The gyroelongated pentagonal birotunda is one of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form. In the illustration to the right, each pentagonal face on the bottom half of the figure is connected by a path of two triangular faces to a pentagonal face above it and to the left. In the figure of opposite chirality (the mirror image of the illustrated figure), each bottom pentagon would be connected to a pentagonal face above it and to the right. The two chiral forms of $2x10(3.5.3.5) 2.10(34.5)$ are not considered different Johnson solids.

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


 * $$A=\left(10\sqrt{3} + 3\sqrt{25+10\sqrt{5}}\right) a^2\approx37.966236883...a^2,$$

and the volume is


 * $$V=\left(\frac{45}{6}+\frac{17}{6}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3\approx20.584813812...a^3.$$