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.

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

3D model of an elongated pentagonal pyramid

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 (${\displaystyle H}$), surface area (${\displaystyle A}$) and volume (${\displaystyle V}$) can be used if all faces are regular, with edge length ${\displaystyle L}$:^[2]

${\displaystyle H=L\cdot \left(1+{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx L\cdot 1.525731112}$

${\displaystyle A=L^{2}\cdot {\frac {20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}\approx L^{2}\cdot 8.88554091}$

${\displaystyle V=L^{3}\cdot \left({\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\right)\approx L^{3}\cdot 2.021980233}$

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

• Elongated pentagonal bipyramid

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

• Weisstein, Eric W., "Johnson solid" ("Elongated pentagonal pyramid") at MathWorld.