| Type: | Johnson |
| Faces: | 4x10 triangles 2+10 pentagons |
| Edges: | 90 |
| Vertices: | 40 |
| Dual: | - |
| Properties: | convex, chiral |
| Net: | Johnson solid 48 net.png |
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.
With edge length a, the surface area is
A=\left(10\sqrt{3}+3\sqrt{25+10\sqrt{5}}\right)a2 ≈ 37.966236883...a2,
and the volume is
| V=\left( | 45 | + |
| 6 |
| 17 | |
| 6 |
\sqrt{5}+
| 5 | |
| 6 |
\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right)a3 ≈ 20.584813812...a3.