Pentagonal cupola

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Pentagonal cupola
Pentagonal cupola.png
J4 - J5 - J6
Faces5 triangles
5 squares
1 pentagon
1 decagon
Vertex configuration10(3.4.10)
Symmetry groupC5v, [5], (*55)
Rotation groupC5, [5]+, (55)
Dual polyhedron-
Pentagonal Cupola.PNG
3D model a pentagonal cupola

In geometry, the pentagonal cupola is one of the Johnson solids (J5). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.

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]


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

The height of the pentagonal cupola is [3]


Related polyhedra[edit]

Dual polyhedron[edit]

The dual of the pentagonal cupola has 10 triangular faces and 5 kite faces:

Dual pentagonal cupola Net of dual 3D model
Dual pentagonal cupola.png Dual pentagonal cupola net.png Pentagonal trapezopyramid.stl

Other convex cupolae[edit]

Family of convex cupolae
n 2 3 4 5 6
Name {2} || t{2} {3} || t{3} {4} || t{4} {5} || t{5} {6} || t{6}
Cupola Triangular prism wedge.png
Digonal cupola
Triangular cupola.png
Triangular cupola
Square cupola.png
Square cupola
Pentagonal cupola.png
Pentagonal cupola
Hexagonal cupola flat.png
Hexagonal cupola
Triangular prism
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png

CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png

CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png

CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png

Crossed pentagrammic cupola[edit]

3D model of a crossed pentagrammic cupola

In geometry, the crossed pentagrammic cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex pentagonal cupola. It can be obtained as a slice of the nonconvex great rhombicosidodecahedron or quasirhombicosidodecahedron, analogously to how the pentagonal cupola may be obtained as a slice of the rhombicosidodecahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is a decagram.

It may be seen as a cupola with a retrograde pentagrammic base, so that the squares and triangles connect across the bases in the opposite way to the pentagrammic cuploid, hence intersecting each other more deeply.


  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 cupola" from Wolfram Alpha. Retrieved April 11, 2020.
  3. ^ Sapiña, R. "Area and volume of the Johnson solid J5". Problemas y ecuaciones (in Spanish). ISSN 2659-9899. Retrieved 2020-07-16.

External links[edit]