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 ) {\displaystyle 10\times (3\times 4\times 10)} 5 × ( 3 × 4 × 5 × 4 ) {\displaystyle 5\times (3\times 4\times 5\times 4)}
Symmetry group	C v {\displaystyle C_{\mathrm {v} }}
Properties	convex, elementary
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.

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 R {\displaystyle R} , and height h {\displaystyle h} , surface area A {\displaystyle A} , and volume V {\displaystyle V} may be applied if all faces are regular with edge length a {\displaystyle a} :^[4] h = 5 − 5 10 a ≈ 0.526 a , R = 11 + 4 5 2 a ≈ 2.233 a , A = 20 + 5 3 + 5 ( 145 + 62 5 ) 4 a 2 ≈ 16.580 a 2 , V = 5 + 4 5 6 a 3 ≈ 2.324 a 3 . {\displaystyle {\begin{aligned}h&={\sqrt {\frac {5-{\sqrt {5}}}{10}}}a&\approx 0.526a,\\R&={\frac {\sqrt {11+4{\sqrt {5}}}}{2}}a&\approx 2.233a,\\A&={\frac {20+5{\sqrt {3}}+{\sqrt {5\left(145+62{\sqrt {5}}\right)}}}{4}}a^{2}&\approx 16.580a^{2},\\V&={\frac {5+4{\sqrt {5}}}{6}}a^{3}&\approx 2.324a^{3}.\end{aligned}}}

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 C 5 v {\displaystyle C_{5\mathrm {v} }} 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 J 20 {\displaystyle J_{20}}
• gyroelongated pentagonal cupola J 24 {\displaystyle J_{24}}
• pentagonal orthobicupola J 30 {\displaystyle J_{30}}
• pentagonal gyrobicupola J 31 {\displaystyle J_{31}}
• pentagonal orthocupolarotunda J 32 {\displaystyle J_{32}}
• pentagonal gyrocupolarotunda J 33 {\displaystyle J_{33}}
• elongated pentagonal orthobicupola J 38 {\displaystyle J_{38}}
• elongated pentagonal gyrobicupola J 39 {\displaystyle J_{39}}
• elongated pentagonal orthocupolarotunda J 40 {\displaystyle J_{40}}
• gyroelongated pentagonal bicupola J 46 {\displaystyle J_{46}}
• gyroelongated pentagonal cupolarotunda J 47 {\displaystyle J_{47}}
• augmented truncated dodecahedron J 68 {\displaystyle J_{68}}
• parabiaugmented truncated dodecahedron J 69 {\displaystyle J_{69}}
• metabiaugmented truncated dodecahedron J 70 {\displaystyle J_{70}}
• triaugmented truncated dodecahedron J 71 {\displaystyle J_{71}}
• gyrate rhombicosidodecahedron J 72 {\displaystyle J_{72}}
• parabigyrate rhombicosidodecahedron J 73 {\displaystyle J_{73}}
• metabigyrate rhombicosidodecahedron J 74 {\displaystyle J_{74}}
• trigyrate rhombicosidodecahedron J 75 {\displaystyle J_{75}}

Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishment^[1]:

• diminished rhombicosidodecahedron J 76 {\displaystyle J_{76}}
• paragyrate diminished rhombicosidodecahedron J 77 {\displaystyle J_{77}}
• metagyrate diminished rhombicosidodecahedron J 78 {\displaystyle J_{78}}
• bigyrate diminished rhombicosidodecahedron J 79 {\displaystyle J_{79}}
• parabidiminished rhombicosidodecahedron J 80 {\displaystyle J_{80}}
• metabidiminished rhombicosidodecahedron J 81 {\displaystyle J_{81}}
• gyrate bidiminished rhombicosidodecahedron J 82 {\displaystyle J_{82}}
• tridiminished rhombicosidodecahedron J 83 {\displaystyle J_{83}}

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