Gyroelongated pentagonal rotunda
| Gyroelongated pentagonal rotunda | |
|---|---|
 | |
| Type | Johnson J24 - J25 - J26 |
| Faces | 4x5+10 triangles 1+5 pentagons 1 decagon |
| Edges | 65 |
| Vertices | 30 |
| Vertex configuration | 2.5(3.5.3.5) 2.5(33.10) 10(34.5) |
| Symmetry group | C5v |
| Dual polyhedron | see above |
| Properties | convex |
| Net | |
 | |
In geometry, the gyroelongated pentagonal rotunda is one of the Johnson solids (J25). As the name suggests, it can be constructed by gyroelongating a pentagonal rotunda (J6) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal birotunda (J48) with one pentagonal rotunda 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}{2}\left( 15\sqrt{3}+\left(5+3\sqrt{5}\right)\sqrt{5+2\sqrt{5}}\right)a^2\approx31.007454303...a^2, }[/math]
and the volume is
- [math]\displaystyle{ V=\left(\frac{45}{12}+\frac{17}{12}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3\approx13.667050844...a^3. }[/math]
Dual polyhedron
The dual of the gyroelongated pentagonal rotunda has 30 faces: 10 pentagons, 10 rhombi, and 10 quadrilaterals.
| Dual gyroelongated pentagonal rotunda | 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.
