In geometry, the pentagonal gyrocupolarotunda is one of the Johnson solids (). Like the pentagonal orthocupolarotunda (), it can be constructed by joining a pentagonal cupola () and a pentagonal rotunda () along their decagonal bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another.
thumb|3D model of a pentagonal gyrocupolarotunda
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:
:<math>V=\frac{5}{12}\left(11+5\sqrt{5}\right)a^3\approx9.24181...a^3</math>
:<math>A= \left(5+\frac{15}{4}\sqrt{3}+\frac{7}{4}\sqrt{25+10\sqrt{5\right) a^2\approx23.5385...a^2</math>