View source for Gyroelongated pentagonal rotunda

{{Infobox polyhedron
|image=gyroelongated_pentagonal rotunda.png
|type=[[Johnson solid|Johnson]]<br>[[Gyroelongated pentagonal cupola|''J''<sub>24</sub>]] -''' J<sub>25</sub>''' - [[Gyrobifastigium|J<sub>26</sub>]]
|faces=4x5+10 [[Triangle|triangle]]s<br>1+5 [[Pentagon|pentagon]]s<br>1 [[Decagon|decagon]]
|edges=65
|vertices=30
|symmetry=''C''<sub>5v</sub>|
|vertex_config=2.5(3.5.3.5)<br>2.5(3<sup>3</sup>.10)<br>10(3<sup>4</sup>.5)
|dual=see above
|properties=[[Convex set|convex]]
|net=Johnson solid 25 net.png
}}

In [[Geometry|geometry]], the '''gyroelongated pentagonal rotunda''' is one of the [[Johnson solid]]s (''J''<sub>25</sub>). As the name suggests, it can be constructed by gyroelongating a [[Pentagonal rotunda|pentagonal rotunda]] (''J''<sub>6</sub>) by attaching a [[Decagonal antiprism|decagonal antiprism]] to its base. It can also be seen as a [[Gyroelongated pentagonal birotunda|gyroelongated pentagonal birotunda]] (''J''<sub>48</sub>) with one pentagonal rotunda removed.

{{Johnson solid}}

== Area and Volume ==
With edge length a, the surface area is

:<math>A=\frac{1}{2}\left( 15\sqrt{3}+\left(5+3\sqrt{5}\right)\sqrt{5+2\sqrt{5}}\right)a^2\approx31.007454303...a^2,</math>

and the volume is

:<math>V=\left(\frac{45}{12}+\frac{17}{12}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3\approx13.667050844...a^3.</math>

== Dual polyhedron ==

The dual of the gyroelongated pentagonal rotunda has 30 faces: 10 pentagons, 10 rhombi, and 10 quadrilaterals.
{| class=wikitable width=320
|- valign=top
!Dual gyroelongated pentagonal rotunda
!Net of dual
|- valign=top
|[[File:Dual gyroelongated pentagonal rotunda.png|160px]]
|[[File:Dual gyroelongated pentagonal rotunda net.png|160px]]
|}

==External links==
* {{mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid| urlname = GyroelongatedPentagonalRotunda  | title = Gyroelongated pentagonal rotunda }}
{{Johnson solids navigator}}
[[Category:Johnson solids]]

{{Sourceattribution|Gyroelongated pentagonal rotunda}}