Pentagonal cupola

Pentagonal cupola
Type	Johnson J4 – J5 – J6
Faces	5 triangles 5 squares 1 pentagon 1 decagon
Edges	25
Vertices	15
Vertex configuration	10(3.4.10) 5(3.4.5.4)
Symmetry group	C5v, [5], (*55)
Rotation group	C5, [5]+, (55)
Dual polyhedron	-
Properties	convex
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

• v
• t
• e

n	2	3	4	5	6
Name	{2} || t{2}	{3} || t{3}	{4} || t{4}	{5} || t{5}	{6} || t{6}
Cupola	Digonal cupola	Triangular cupola	Square cupola	Pentagonal cupola	Hexagonal cupola (Flat)
Related uniform polyhedra	Triangular prism	Cubocta- hedron	Rhombi- cubocta- hedron	Rhomb- icosidodeca- hedron	Rhombi- trihexagonal 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.

External links

• Eric W. Weisstein, Pentagonal cupola ( Johnson solid ) at MathWorld.