Pentagonal cupola | |
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Type | Johnson J4 – J5 – J6 |
Faces | 5 triangles 5 squares 1 pentagon 1 decagon |
Edges | 25 |
Symmetry group | |
Properties | convex, elementary |
Net | |
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The pentagonal cupola's faces are five equilateral triangles, five squares, one regular pentagon, and one regular decagon. [1] It has the property of convexity and regular polygonal faces, from which it is classified as the fifth Johnson solid. [2] This cupola cannot be sliced by a plane without cutting within a face, so it is an elementary polyhedron. [3]
The following formulae for circumradius , and height , surface area , and volume may be applied if all faces are regular with edge length : [4]
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. [3]
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. [5] [6] Some of the Johnson solids with such constructions are: elongated pentagonal cupola , gyroelongated pentagonal cupola , pentagonal orthobicupola , pentagonal gyrobicupola , pentagonal orthocupolarotunda , pentagonal gyrocupolarotunda , elongated pentagonal orthobicupola , elongated pentagonal gyrobicupola , elongated pentagonal orthocupolarotunda , gyroelongated pentagonal bicupola , gyroelongated pentagonal cupolarotunda , augmented truncated dodecahedron , parabiaugmented truncated dodecahedron , metabiaugmented truncated dodecahedron , triaugmented truncated dodecahedron , gyrate rhombicosidodecahedron , parabigyrate rhombicosidodecahedron , metabigyrate rhombicosidodecahedron , and trigyrate rhombicosidodecahedron . Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishment: diminished rhombicosidodecahedron , paragyrate diminished rhombicosidodecahedron , metagyrate diminished rhombicosidodecahedron , bigyrate diminished rhombicosidodecahedron , parabidiminished rhombicosidodecahedron , metabidiminished rhombicosidodecahedron , gyrate bidiminished rhombicosidodecahedron , and tridiminished rhombicosidodecahedron . [1]
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