Elongated pentagonal pyramid
| Elongated pentagonal pyramid | |
|---|---|
 | |
| Type | Johnson J8 – J9 – J10 |
| Faces | 5 triangles 5 squares 1 pentagon |
| Edges | 20 |
| Vertices | 11 |
| Vertex configuration | 5(42.5) 5(32.42) 1(35) |
| Symmetry group | C5v, [5], (*55) |
| Rotation group | C5, [5]+, (55) |
| Dual polyhedron | self |
| Properties | convex |
| Net | |
 | |
File:J9 elongated pentagonal pyramid.stl
In geometry, the elongated pentagonal pyramid is one of the Johnson solids (J9). As the name suggests, it can be constructed by elongating a pentagonal pyramid (J2) by attaching a pentagonal prism to its base.
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 the height ([math]\displaystyle{ H }[/math]), surface area ([math]\displaystyle{ A }[/math]) and volume ([math]\displaystyle{ V }[/math]) can be used if all faces are regular, with edge length [math]\displaystyle{ L }[/math]:[2]
- [math]\displaystyle{ H = L\cdot \left( 1 + \sqrt{\frac{5 - \sqrt{5}}{10}}\right) \approx L\cdot 1.525731112 }[/math]
- [math]\displaystyle{ A = L^2 \cdot \frac{20 + 5\sqrt{3} + \sqrt{25 + 10\sqrt{5}}}{4} \approx L^2\cdot 8.88554091 }[/math]
- [math]\displaystyle{ V = L^3 \cdot \left( \frac{5 + \sqrt{5} + 6\sqrt{25 + 10\sqrt{5}}}{24} \right) \approx L^3\cdot 2.021980233 }[/math]
Dual polyhedron
The dual of the elongated pentagonal pyramid has 11 faces: 5 triangular, 1 pentagonal and 5 trapezoidal. It is topologically identical to the Johnson solid.
| Dual elongated pentagonal pyramid | Net of dual |
|---|---|
|
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See also
References
- ↑ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, doi:10.4153/cjm-1966-021-8.
- ↑ Sapiña, R.. "Area and volume of the Johnson solid J9" (in es). Problemas y ecuaciones. ISSN 2659-9899. https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J9/calculadora-area-volumen-formulas.html. Retrieved 2020-08-30.
External links
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