Pentagonal gyrobicupola
Johnson solid
The pentagonal gyrobicupola is a polyhedron that is constructed by attaching two pentagonal cupolas base-to-base, each of its cupolas is twisted at 36°. It is an example of a Johnson solid and a composite polyhedron.
Construction
The pentagonal gyrobicupola is a composite polyhedron: it is constructed by attaching two pentagonal cupolas base-to-base. This construction is similar to the pentagonal orthobicupola; the difference is that one of the cupolas in the pentagonal gyrobicupola is twisted at 36°, as suggested by the prefix gyro-. The resulting polyhedron has the same faces as the pentagonal orthobicupola does: those cupolas cover their decagonal bases, replacing them with ten equilateral triangles, ten squares, and two regular pentagons. A convex polyhedron in which all of its faces are regular polygons is the Johnson solid. The pentagonal gyrobicupola has these, enumerating it as the thirty-first Johnson solid
J
31
{\displaystyle J_{31}}
.
Properties
The surface area of a pentagonal gyrobicupola
A
{\displaystyle A}
is the sum of its faces' area, and its volume
V
{\displaystyle V}
is twice the volume of a pentagonal cupola:
A
=
20
+
100
+
10
5
+
10
75
+
30
5
2
a
2
≈
17.771
a
2
,
V
=
5
+
4
5
3
a
3
≈
4.648
a
3
.
{\displaystyle {\begin{aligned}A&={\frac {20+{\sqrt {100+10{\sqrt {5}}+10{\sqrt {75+30{\sqrt {5}}}}}}}{2}}a^{2}\approx 17.771a^{2},\\V&={\frac {5+4{\sqrt {5}}}{3}}a^{3}\approx 4.648a^{3}.\end{aligned}}}
The pentagonal gyrobicupola has a three-dimensional symmetry group, the antiprismatic symmetry of
D
5
d
{\displaystyle D_{5d}}
. Its dihedral angles (i.e., the angle between two adjacent polygonal faces) are as follows:
the angle between a pentagon and a square is 159.09°.
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