Elongated pentagonal orthocupolarotunda

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Short description: 40th Johnson solid
Elongated pentagonal orthocupolarotunda
Elongated pentagonal orthocupolarotunda.png
TypeJohnson
J39J40J41
Faces3x5 triangles
3x5 squares
2+5 pentagons
Edges70
Vertices35
Vertex configuration10(3.43)
10(3.42.5)
5(3.4.5.4)
2.5(3.5.3.5)
Symmetry groupC5v
Dual polyhedron-
Propertiesconvex
Net
Johnson solid 40 net.png

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

  1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, doi:10.4153/cjm-1966-021-8 .
  2. Stephen Wolfram, "Elongated pentagonal orthocupolarotunda" from Wolfram Alpha. Retrieved July 25, 2010.

External links