Gyroelongated pentagonal cupola
| Gyroelongated pentagonal cupola | |
|---|---|
 | |
| Type | Johnson J23 - J24 - J25 |
| Faces | 3x5+10 triangles 5 squares 1 pentagon 1 decagon |
| Edges | 55 |
| Vertices | 25 |
| Vertex configuration | 5(3.4.5.4) 2.5(33.10) 10(34.4) |
| Symmetry group | C5v |
| Dual polyhedron | - |
| Properties | convex |
| Net | |
 | |
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.
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
Area and Volume
With edge length a, the surface area is
- [math]\displaystyle{ A=\frac{1}{4}\left( 20+25\sqrt{3}+\left(10+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}\right)a^2\approx25.240003791...a^2, }[/math]
and the volume is
- [math]\displaystyle{ V=\left(\frac{5}{6}+\frac{2}{3}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3\approx 9.073333194...a^3. }[/math]
Dual polyhedron
The dual of the gyroelongated pentagonal cupola has 25 faces: 10 kites, 5 rhombi, and 10 pentagons.
| Dual gyroelongated pentagonal cupola | Net of dual |
|---|---|
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External links
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- ↑ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, doi:10.4153/cjm-1966-021-8.
