Elongated pentagonal cupola
| Elongated pentagonal cupola | |
|---|---|
 | |
| Type | Johnson J19 – J20 – J21 |
| Faces | 5 triangles 15 squares 1 pentagon 1 decagon |
| Edges | 45 |
| Vertices | 25 |
| Vertex configuration | 10(42.10) 10(3.43) 5(3.4.5.4) |
| Symmetry group | C5v |
| Dual polyhedron | - |
| Properties | convex |
| Net | |
 | |
In geometry, the elongated pentagonal cupola is one of the Johnson solids (J20). As the name suggests, it can be constructed by elongating a pentagonal cupola (J5) by attaching a decagonal prism to its base. The solid can also be seen as an elongated pentagonal orthobicupola (J38) with its "lid" (another 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]
Formulas
The following formulas for the volume and surface area can be used if all faces are regular, with edge length a:[2]
- [math]\displaystyle{ V=\left(\frac{1}{6}\left(5+4\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)\right)a^3\approx10.0183...a^3 }[/math]
- [math]\displaystyle{ A=\left(\frac{1}{4}\left(60+\sqrt{10\left(80+31\sqrt{5}+\sqrt{2175+930\sqrt{5}}\right)}\right)\right)a^2\approx26.5797...a^2 }[/math]
Dual polyhedron
The dual of the elongated pentagonal cupola has 25 faces: 10 isosceles triangles, 5 kites, and 10 quadrilaterals.
| Dual elongated pentagonal cupola | Net of dual |
|---|---|
|
|
References
- ↑ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, doi:10.4153/cjm-1966-021-8.
- ↑ Stephen Wolfram, "Elongated pentagonal cupola" from Wolfram Alpha. Retrieved July 22, 2010.
External links
 |  |
