Gyroelongated pentagonal birotunda

In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids (Template:Math). As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda (either Template:Math 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 Template:Math are not considered different Johnson solids. File:J48 gyroelongated pentagonal birotunda.stl

Area and volume

With edge length a, the surface area is

$A=(10\sqrt{3}+3\sqrt{25+10\sqrt{5}})a^2\approx \mathrm{37.966236883...}{a}^2,$

and the volume is

$V=(\frac{45}{6}+\frac{17}{6}\sqrt{5}+\frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2})a^3$ $\approx \mathrm{20.584813812...}{a}^3.$

See also

• Birotunda

External links

• Template:MathWorld
  • Template:MathWorld