Gyroelongated pentagonal bicupola

In geometry, the gyroelongated pentagonal bicupola is one of the Johnson solids ($J45 – J46 – J47$). As the name suggests, it can be constructed by gyroelongating a pentagonal bicupola ($D5$ or $10(3.4.5.4) 2.10(34.4)$) by inserting a decagonal antiprism between its congruent halves.

The gyroelongated pentagonal bicupola 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 square face on the bottom half of the figure is connected by a path of two triangular faces to a square face above it and to the right. In the figure of opposite chirality (the mirror image of the illustrated figure), each bottom square would be connected to a square face above it and to the left. The two chiral forms of $J46$ are not considered different Johnson solids.

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


 * $$A=\frac{1}{2}\left(20+15\sqrt{3}+\sqrt{25+10\sqrt{5}}\right)a^2\approx26.431335858...a^2,$$

and the volume is


 * $$V=\left(\frac{5}{3}+\frac{4}{3}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3\approx11.397378512...a^3.$$