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.

3D model of an elongated pentagonal rotunda

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a:^[1]

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

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

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

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