Elongated pentagonal gyrocupolarotunda

In geometry, the elongated pentagonal gyrocupolarotunda is one of the Johnson solids (Template:Math). As the name suggests, it can be constructed by elongating a pentagonal gyrocupolarotunda (Template:Math) by inserting a decagonal prism between its halves. Rotating either the pentagonal cupola (Template:Math) or the pentagonal rotunda (Template:Math) through 36 degrees before inserting the prism yields an elongated pentagonal orthocupolarotunda (Template:Math).

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a:^[1]

V = \frac{5}{12} (11 + 5 \sqrt{5} + 6 \sqrt{5 + 2 \sqrt{5}}) a^{3} \approx 16.936 . . . a^{3}

A = \frac{1}{4} (60 + \sqrt{10 (190 + 49 \sqrt{5} + 21 \sqrt{75 + 30 \sqrt{5}})}) a^{2} \approx 33.5385 . . . a^{2}