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.

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 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.Template:R 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 $J_{31}$.Template:R

Properties

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

External links

• Template:MathWorld2