Elongated pentagonal rotunda

The elongated pentagonal rotunda, or epro, is one of the 92 Johnson solids (J21). It consists of 5+5 triangles, 5+5 squares, 1+5 pentagons, and 1 decagon. It can be constructed by attaching a decagonal prism to the decagonal base of the pentagonal rotunda.

If a second rotunda is attached to the other decagonal base of the prism in the same orientation, the result is the elongated pentagonal orthobirotunda. If the second rotunda is rotated 36º instead, the result is the elongated pentagonal gyrobirotunda.

Vertex coordinates
An elongated pentagonal rotunda of edge length 1 has the following vertices:
 * $$\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,±\frac12\right),$$
 * $$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right),$$
 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right),$$
 * $$\left(0,\,-\sqrt{\frac{5+2\sqrt5}{5}},\,\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{25+11\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,-\sqrt{\frac{5+\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right),$$