# Pentagonal cupola

Pentagonal cupola
TypeJohnson
J4J5J6
Faces5 triangles
5 squares
1 pentagon
1 decagon
Edges25
Vertices15
Vertex configuration10(3.4.10)
5(3.4.5.4)
Symmetry groupC5v, [5], (*55)
Rotation groupC5, [5]+, (55)
Dual polyhedron-
Propertiesconvex
Net
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]

## Formulae

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

${\displaystyle V=\left({\frac {1}{6}}\left(5+4{\sqrt {5}}\right)\right)a^{3}\approx 2.32405a^{3},}$
${\displaystyle A=\left({\frac {1}{4}}\left(20+5{\sqrt {3}}+{\sqrt {5\left(145+62{\sqrt {5}}\right)}}\right)\right)a^{2}\approx 16.57975a^{2},}$
${\displaystyle R=\left({\frac {1}{2}}{\sqrt {11+4{\sqrt {5}}}}\right)a\approx 2.23295a.}$

The height of the pentagonal cupola is [3]

${\displaystyle h={\sqrt {\frac {5-{\sqrt {5}}}{10}}}a\approx 0.52573a}$.

## Related polyhedra

### Dual polyhedron

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

Dual pentagonal cupola Net of dual 3D model

### Other convex cupolae

Family of convex cupolae
n 2 3 4 5 6 7 8
Schläfli symbol {2} || t{2} {3} || t{3} {4} || t{4} {5} || t{5} {6} || t{6} {7} || t{7} {8} || t{8}
Cupola
Digonal cupola

Triangular cupola

Square cupola

Pentagonal cupola

Hexagonal cupola
(Flat)

Heptagonal cupola
(Non-regular face)

Octagonal cupola
(Non-regular face)
Related
uniform
polyhedra
Rhombohedron
Cubocta-
hedron

Rhombi-
cubocta-
hedron

Rhomb-
icosidodeca-
hedron

Rhombi-
trihexagonal
tiling

Rhombi-
triheptagonal
tiling
Rhombi-
trioctagonal
tiling

### Crossed pentagrammic cupola

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.

## 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. ^ 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.