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

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]

${\displaystyle V=\left({\frac {1}{6))\left(5+4{\sqrt {5))\right)\right)a^{3}\approx 2.32405a^{3},}$

${\displaystyle A=\left({\frac {1}{4))\left(20+5{\sqrt {3))+{\sqrt {5\left(145+62{\sqrt {5))\right)))\right)\right)a^{2}\approx 16.57975a^{2},}$

${\displaystyle R=\left({\frac {1}{2)){\sqrt {11+4{\sqrt {5))))\right)a\approx 2.23295a.}$

The height of the pentagonal cupola is ^[3]

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

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

Dual pentagonal cupola	Net of dual	3D model

Family of convex cupolae

• v
• t
• e

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	Cuboctahedron	Rhombicuboctahedron	Rhombicosidodecahedron	Rhombitrihexagonal tiling	Rhombitriheptagonal tiling	Rhombitrioctagonal tiling

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.

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