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

In geometry, the pentagonal cupola is one of the Johnson solids (J₅). It can be obtained as a slice of the rhombicosidodecahedron.

The 92 Johnson solids were named and described by Norman Johnson in 1966.

The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.

Formulae

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

$V=\left(\frac{1}{6}\left(5+4\sqrt{5}\right)\right)a^3\approx2.32405...a^3$

$A=\left(\frac{1}{4}\left(20+5\sqrt{3}+\sqrt{5(145+62\sqrt{5})}\right)\right)a^2=\left(\frac{1}{4}\left(20+\sqrt{10\left(80+31\sqrt{5}+\sqrt{15(145+62\sqrt{5})}\right)}\right)\right)a^2\approx16.5797...a^2$

$C=\left(\frac{1}{2}\sqrt{11+4\sqrt{5}}\right)a\approx2.23295...a$

Related polyhedra

Dual polyhedron

The dual of the pentagonal cupola has 20 triangular faces:

Dual pentagonal cupola	Net of dual

Family of cupolae

2	3	4	5	6
Digonal cupola	Triangular cupola	Square cupola	Pentagonal cupola	Hexagonal cupola (Flat)

External links

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