Elongated pentagonal cupola

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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:<ref>Stephen Wolfram, "Elongated pentagonal cupola" from Wolfram Alpha. Retrieved July 22, 2010.</ref>

<math>V=\left(\frac{1}{6}\left(5+4\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)\right)a^3\approx10.0183...a^3</math>

<math>A=\left(\frac{1}{4}\left(60+\sqrt{10\left(80+31\sqrt{5}+\sqrt{2175+930\sqrt{5}}\right)}\right)\right)a^2\approx26.5797...a^2</math>

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
File:Dual elongated pentagonal cupola.png	File:Dual elongated pentagonal cupola net.png

External links

• Template:Mathworld2