In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda (either 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 are not considered different Johnson solids.
thumb|3D model of a gyroelongated pentagonal birotunda
Area and volume
With edge length a, the surface area is
:<math>A=\left(10\sqrt{3} + 3\sqrt{25+10\sqrt{5\right) a^2\approx37.966236883...a^2,</math>
and the volume is
:<math>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</math> <math>\approx20.584813812...a^3.</math>
See also
- Birotunda