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.

Properties

The pentagonal cupola's faces are five equilateral triangles, five squares, one regular pentagon, and one regular decagon. It has the property of convexity and regular polygonal faces, from which it is classified as the fifth Johnson solid. This cupola produces two or more regular polyhedrons by slicing it with a plane, an elementary polyhedron's example. The following formulae for circumradius R, and height h, surface area A, and volume V may be applied if all faces are regular with edge length a: It has an axis of symmetry passing through the center of both top and base, which is symmetrical by rotating around it at one-, two-, three-, and four-fifth of a full-turn angle. It is also mirror-symmetric relative to any perpendicular plane passing through a bisector of the hexagonal base. Therefore, it has pyramidal symmetry, the cyclic group of order ten.

Related polyhedron

The pentagonal cupola can be applied to construct a polyhedron. A construction that involves the attachment of its base to another polyhedron is known as augmentation; attaching it to prisms or antiprisms is known as elongation or gyroelongation. Some of the Johnson solids with such constructions are: elongated pentagonal cupola J_{20}, gyroelongated pentagonal cupola J_{24}, pentagonal orthobicupola J_{30}, pentagonal gyrobicupola J_{31}, pentagonal orthocupolarotunda J_{32}, pentagonal gyrocupolarotunda J_{33}, elongated pentagonal orthobicupola J_{38}, elongated pentagonal gyrobicupola J_{39}, elongated pentagonal orthocupolarotunda J_{40}, gyroelongated pentagonal bicupola J_{46}, gyroelongated pentagonal cupolarotunda J_{47}, augmented truncated dodecahedron J_{68}, parabiaugmented truncated dodecahedron J_{69}, metabiaugmented truncated dodecahedron J_{70}, triaugmented truncated dodecahedron J_{71}, gyrate rhombicosidodecahedron J_{72}, parabigyrate rhombicosidodecahedron J_{73}, metabigyrate rhombicosidodecahedron J_{74}, and trigyrate rhombicosidodecahedron J_{75}. Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishment: diminished rhombicosidodecahedron J_{76}, paragyrate diminished rhombicosidodecahedron J_{77}, metagyrate diminished rhombicosidodecahedron J_{78}, bigyrate diminished rhombicosidodecahedron J_{79}, parabidiminished rhombicosidodecahedron J_{80}, metabidiminished rhombicosidodecahedron J_{81}, gyrate bidiminished rhombicosidodecahedron J_{82}, and tridiminished rhombicosidodecahedron J_{83}.