| Type: | Johnson |
| Faces: | 10 triangles 1+5 pentagons 1 decagon |
| Edges: | 35 |
| Vertices: | 20 |
| Dual: | - |
| Properties: | convex |
| Net: | Pentagonal Rotunda Net.svg |
In geometry, the pentagonal rotunda is one of the Johnson solids . It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces.
The following formulae for volume, surface area, circumradius, and height are valid if all faces are regular, with edge length a:
| V=\left( | 1 |
| 12 |
\left(45+17\sqrt{5}\right)\right)a3 ≈ 6.91776...a3
| \begin{align} A&=\left( | 1 |
| 2 |
\sqrt{5\left(145+58\sqrt{5}+2\sqrt{30\left(65+29\sqrt{5}\right)}\right)}\right)a2\\ &=\left(
| 1 | |
| 2 |
\left(5\sqrt{3}+\sqrt{10\left(65+29\sqrt{5}\right)}\right)\right)a2 ≈ 22.3472...a2 \end{align}
| R=\left( | 1 |
| 2 |
\left(1+\sqrt{5}\right)\right)a ≈ 1.61803...a
| H=\left(\sqrt{1+ | 2 |
| \sqrt{5 |
The dual of the pentagonal rotunda has 20 faces: 10 triangular, 5 rhombic, and 5 kites.