Elongated pentagonal orthocupolarotunda
| Elongated pentagonal orthocupolarotunda | |
|---|---|
 | |
| Type | Johnson J39 – J40 – J41 |
| Faces | 3x5 triangles 3x5 squares 2+5 pentagons |
| Edges | 70 |
| Vertices | 35 |
| Vertex configuration | 10(3.43) 10(3.42.5) 5(3.4.5.4) 2.5(3.5.3.5) |
| Symmetry group | C5v |
| Dual polyhedron | - |
| Properties | convex |
| Net | |
 | |
In geometry, the elongated pentagonal orthocupolarotunda is one of the Johnson solids (J40). As the name suggests, it can be constructed by elongating a pentagonal orthocupolarotunda (J32) by inserting a decagonal prism between its halves. Rotating either the cupola or the rotunda through 36 degrees before inserting the prism yields an elongated pentagonal gyrocupolarotunda (J41).
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]
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:[2]
- [math]\displaystyle{ V=\frac{5}{12}\left(11+5\sqrt{5}+6\sqrt{5+2\sqrt{5}}\right)a^3\approx16.936...a^3 }[/math]
- [math]\displaystyle{ A=\frac{1}{4}\left(60+\sqrt{10\left(190+49\sqrt{5}+21\sqrt{75+30\sqrt{5}}\right)}\right)a^2\approx33.5385...a^2 }[/math]
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 orthocupolarotunda" from Wolfram Alpha. Retrieved July 25, 2010.
External links
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