- Source: Pentagonal cupola
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
{\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}
:
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.
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
{\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}}
, and trigyrate rhombicosidodecahedron
J
75
{\displaystyle J_{75}}
. Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishment: 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}}
, and tridiminished rhombicosidodecahedron
J
83
{\displaystyle J_{83}}
.
References
External links
Weisstein, Eric W., "Pentagonal cupola" ("Johnson solid") at MathWorld.
Kata Kunci Pencarian:
- Daftar bentuk matematika
- Pentagonal cupola
- Elongated pentagonal cupola
- Gyroelongated pentagonal cupola
- Cupola (geometry)
- Pentagonal orthocupolarotunda
- Pentagonal gyrocupolarotunda
- Pentagonal gyrobicupola
- Elongated pentagonal gyrocupolarotunda
- Johnson solid
- List of polygons, polyhedra and polytopes
