Pentagonal orthocupolarotunda
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In geometry, the pentagonal orthocupolarotunda is one of the Johnson solids (Template:Math). As the name suggests, it can be constructed by joining a pentagonal cupola (Template:Math) and a pentagonal rotunda (Template:Math) along their decagonal bases, matching the pentagonal faces. A 36-degree rotation of one of the halves before the joining yields a pentagonal gyrocupolarotunda (Template:Math).
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:<ref>Stephen Wolfram, "Pentagonal orthocupolarotunda" from Wolfram Alpha. Retrieved July 24, 2010.</ref>
- <math>V=\frac{5}{12}\left(11+5\sqrt{5}\right)a^3\approx9.24181...a^3</math>
- <math>A=\left(5+\frac{1}{4}\sqrt{1900+490\sqrt{5}+210\sqrt{75+30\sqrt{5}}}\right)a^2\approx23.5385...a^2</math>