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]

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