Elongated pentagonal cupola

The elongated pentagonal cupola is one of the 92 Johnson solids (J20). It consists of 5 triangles, 5+5+5 squares, 1 pentagon, and 1 decagon. It can be constructed by attaching a decagonal prism to the decagonal base of the pentagonal cupola.

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

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