Pentagonal gyrobicupola
Template:Short description {{#invoke:Infobox|infobox}}Template:Template other </math>
| vertex_config = <math> 10 \times (3 \times 4 \times 3 \times 4) </math>
<math> 10 \times (3 \times 4 \times 5 \times 4) </math> | properties = convex, composite | net = Johnson solid 31 net.png
}}
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.Template:R 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 <math> J_{31} </math>.Template:R
Properties
The surface area of a pentagonal gyrobicupola <math> A </math> is the sum of its faces' area, and its volume <math> V </math> is twice the volume of a pentagonal cupola:Template:R <math display="block"> \begin{align}
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{align} </math>
The dihedral angles of a pentagonal gyrobicupola are as follows:Template:R
- 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°.