| Type: | Johnson J23 - J24 - J25 |
| Faces: | 3x5+10 triangles 5 squares 1 pentagon 1 decagon |
| Edges: | 55 |
| Vertices: | 25 |
| Symmetry: | C5v |
| Vertex Config: | 5(3.4.5.4) 2.5(33.10) 10(34.4) |
| Dual: | - |
| Properties: | convex |
| Net: | Johnson solid 24 net.png |
In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (J24). As the name suggests, it can be constructed by gyroelongating a pentagonal cupola (J5) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal bicupola (J46) with one pentagonal cupola removed.
With edge length a, the surface area is
| A= | 1 |
| 4 |
\left(20+25\sqrt{3}+\left(10+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}\right)a2 ≈ 25.240003791...a2,
and the volume is
| V=\left( | 5 | + |
| 6 |
| 2 | |
| 3 |
\sqrt{5}+
| 5 | |
| 6 |
\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right)a3 ≈ 9.073333194...a3.
The dual of the gyroelongated pentagonal cupola has 25 faces: 10 kites, 5 rhombi, and 10 pentagons.