Elongated pentagonal cupola

In geometry, the elongated pentagonal cupola is one of the Johnson solids (Template:Math). As the name suggests, it can be constructed by elongating a pentagonal cupola (Template:Math) by attaching a decagonal prism to its base. The solid can also be seen as an elongated pentagonal orthobicupola (Template:Math) with its "lid" (another pentagonal cupola) removed.

Formulas

The following formulas for the volume and surface area can be used if all faces are regular, with edge length a:

$V=\left(\frac{1}{6}(5+4\sqrt{5}+15\sqrt{5+2\sqrt{5}})\right)a^3\approx 10.0183...a^3$

$A=\left(\frac{1}{4}(60+\sqrt{10(80+31\sqrt{5}+\sqrt{2175+930\sqrt{5}})})\right)a^2\approx 26.5797...a^2$

Dual polyhedron

The dual of the elongated pentagonal cupola has 25 faces: 10 isosceles triangles, 5 kites, and 10 quadrilaterals.

Dual elongated pentagonal cupola	Net of dual

External links

• Template:Mathworld2