Elongated pentagonal rotunda

In geometry, the elongated pentagonal rotunda is one of the Johnson solids (J₂₁). As the name suggests, it can be constructed by elongating a pentagonal rotunda (J₆) by attaching a decagonal prism to its base. It can also be seen as an elongated pentagonal orthobirotunda (J₄₂) with one pentagonal rotunda removed.

Template:Johnson solid File:J21 elongated pentagonal rotunda.stl

Formulae

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

$V=\frac{1}{12}(45+17\sqrt{5}+30\sqrt{5+2\sqrt{5}})a^3\approx \mathrm{14.612...}{a}^3$

$A=\frac{1}{2}(20+\sqrt{5(145+58\sqrt{5}+2\sqrt{30(65+29\sqrt{5})})})a^2\approx \mathrm{32.3472...}{a}^2$

Dual polyhedron

The dual of the elongated pentagonal rotunda has 30 faces: 10 isosceles triangles, 10 rhombi, and 10 quadrilaterals.

Dual elongated pentagonal rotunda	Net of dual

External links

• Template:Mathworld2