Pentagonal orthobicupola

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Pentagonal orthobicupola
Pentagonal orthobicupola.png
Type Johnson
J29 - J30 - J31
Faces 10 triangles
10 squares
2 pentagons
Edges 40
Vertices 20
Vertex configuration 10(32.42)
Symmetry group D5h
Dual polyhedron -
Properties convex
Johnson solid 30 net.png

In geometry, the pentagonal orthobicupola is one of the Johnson solids (J30). As the name suggests, it can be constructed by joining two pentagonal cupolae (J5) along their decagonal bases, matching like faces. A 36-degree rotation of one cupola before the joining yields a pentagonal gyrobicupola (J31).

The pentagonal orthobicupola is the third in an infinite set of orthobicupolae.

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]


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


  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, "Pentagonal orthobicupola" from Wolfram Alpha. Retrieved July 23, 2010.

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