Elongated pentagonal pyramid



In geometry, the elongated pentagonal pyramid is one of the Johnson solids ($J8 – J9 – J10$). As the name suggests, it can be constructed by elongating a pentagonal pyramid ($C5v, [5], (*55)$) by attaching a pentagonal prism to its base.

Formulae
The following formulae for the height ($$H$$), surface area ($$A$$) and volume ($$V$$) can be used if all faces are regular, with edge length $$L$$:
 * $$H = L\cdot \left( 1 + \sqrt{\frac{5 - \sqrt{5}}{10}}\right) \approx L\cdot 1.525731112$$
 * $$A = L^2 \cdot \frac{20 + 5\sqrt{3} + \sqrt{25 + 10\sqrt{5}}}{4} \approx L^2\cdot 8.88554091$$
 * $$V = L^3 \cdot \left(  \frac{5 + \sqrt{5} + 6\sqrt{25 + 10\sqrt{5}}}{24} \right) \approx L^3\cdot 2.021980233$$

Dual polyhedron
The dual of the elongated pentagonal pyramid has 11 faces: 5 triangular, 1 pentagonal and 5 trapezoidal. It is topologically identical to the Johnson solid.