Pentagonal gyrobicupola

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.

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.^[1] 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 ${\displaystyle J_{31}}$.^[2]

The surface area of a pentagonal gyrobicupola ${\displaystyle A}$ is the sum of its faces' area, and its volume ${\displaystyle V}$ is twice the volume of a pentagonal cupola:^[1]$${\displaystyle {\begin{aligned}A&={\frac {20+{\sqrt {100+10{\sqrt {5}}+10{\sqrt {75+30{\sqrt {5}}}}}}}{2}}a^{2}\approx 17.771a^{2},\\V&={\frac {5+4{\sqrt {5}}}{3}}a^{3}\approx 4.648a^{3}.\end{aligned}}}$$

The pentagonal gyrobicupola has a three-dimensional symmetry group, the antiprismatic symmetry of ${\displaystyle D_{5d}}$. Its dihedral angles (i.e., the angle between two adjacent polygonal faces) are as follows:^[3]

• the angle between a pentagon and a square is 159.09°.
• the angle between a square and a triangle, within one cupola, is 148.28°;
• the dihedral angle at the plane joining the two cupolas is the sum of the dihedral angle between square-to-decagon and triangle-to-decagon, 69.09°.

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