Pentagonal gyrobicupola

Pentagonal gyrobicupola
Type	Bicupola, Johnson J30 – J31 – J32
Faces	10 triangles 10 squares 2 pentagons
Edges	40
Vertices	20
Vertex configuration	${\displaystyle 10\times (3\times 4\times 3\times 4)}$ ${\displaystyle 10\times (3\times 4\times 5\times 4)}$
Symmetry group	${\displaystyle D_{5\mathrm {d} }}$
Properties	convex, composite
Net

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 rotundas base-to-base. This construction is similar to the pentagonal orthobicupola; the difference is that one of 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 it with eight equilateral triangles, eight 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 such these, enumerating it as the thirty-first Johnson solid ${\displaystyle J_{31}}$.^[2]

Because it has a similar construction as the pentagonal orthobicupola, the surface area of a pentagonal gyrobicupola ${\displaystyle A}$ is the sum of polygonal faces' area, and its volume ${\displaystyle V}$ is twice the volume of a pentagonal cupola for which slicing it into those:^[1] $${\displaystyle {\begin{aligned}A&={\frac {20+{\sqrt {100+10{\sqrt {5}}+10{\sqrt {75+30{\sqrt {5}}}}}}}{2}}a^{2}\approx 17.771a^{2},\\V&={\frac {5+4{\sqrt {5}}}{3}}a^{3}\approx 4.648a^{3}.\end{aligned}}}$$

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