Elongated pentagonal cupola

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Elongated pentagonal cupola
Elongated pentagonal cupola.png
TypeJohnson
J19 - J20 - J21
Faces5 triangles
15 squares
1 pentagon
1 decagon
Edges45
Vertices25
Vertex configuration10(42.10)
10(3.43)
5(3.4.5.4)
Symmetry groupC5v
Dual polyhedron-
Propertiesconvex
Net
Johnson solid 20 net.png

In geometry, the elongated pentagonal cupola is one of the Johnson solids (J20). As the name suggests, it can be constructed by elongating a pentagonal cupola (J5) by attaching a decagonal prism to its base. The solid can also be seen as an elongated pentagonal orthobicupola (J38) with its "lid" (another pentagonal cupola) removed.

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (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]

Formulas[edit]

The following formulas for the volume and surface area can be used if all faces are regular, with edge length a:[2]

Dual polyhedron[edit]

The dual of the elongated pentagonal cupola has 25 faces: 10 isosceles triangles, 5 kites, and 10 quadrilaterals.

Dual elongated pentagonal cupola Net of dual
Dual elongated pentagonal cupola.png Dual elongated pentagonal cupola net.png

References[edit]

  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. ^ Stephen Wolfram, "Elongated pentagonal cupola" from Wolfram Alpha. Retrieved July 22, 2010.

External links[edit]