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Elongated pentagonal pyramid

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9th Johnson solid (11 faces)
Elongated pentagonal pyramid
TypeJohnson
J8J9J10
Faces5 triangles
5 squares
1 pentagon
Edges20
Vertices11
Vertex configuration5(4.5)
5(3.4)
1(3)
Symmetry groupC5v, , (*55)
Rotation groupC5, , (55)
Dual polyhedronself
Propertiesconvex
Net
3D model of an elongated pentagonal pyramid

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.

Formulae

The following formulae for the height ( H {\displaystyle H} ), surface area ( A {\displaystyle A} ) and volume ( V {\displaystyle V} ) can be used if all faces are regular, with edge length L {\displaystyle L} :

H = L ( 1 + 5 5 10 ) L 1.525731112 {\displaystyle H=L\cdot \left(1+{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx L\cdot 1.525731112}
A = L 2 20 + 5 3 + 25 + 10 5 4 L 2 8.88554091 {\displaystyle A=L^{2}\cdot {\frac {20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}\approx L^{2}\cdot 8.88554091}
V = L 3 ( 5 + 5 + 6 25 + 10 5 24 ) L 3 2.021980233 {\displaystyle V=L^{3}\cdot \left({\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\right)\approx L^{3}\cdot 2.021980233}

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

See also

References

  1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
  2. Sapiña, R. "Area and volume of the Johnson solid J9". Problemas y ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-08-30.

External links

Johnson solids
Pyramids, cupolae and rotundae
Modified pyramids
Modified cupolae and rotundae
Augmented prisms
Modified Platonic solids
Modified Archimedean solids
Other elementary solids
(See also List of Johnson solids, a sortable table)
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