View source for Elongated pentagonal gyrobirotunda

{{Short description|43rd Johnson solid}}
{{Infobox polyhedron
|image=elongated_pentagonal_gyrobirotunda.png
|type=[[Johnson solid|Johnson]]<br>{{math|[[Elongated pentagonal orthobirotunda|''J''{{sub|42}}]] – '''''J''{{sub|43}}''' – [[Gyroelongated triangular bicupola|''J''{{sub|44}}]]}}
|faces=10+10 [[Triangle|triangle]]s<br>10 squares<br>2+10 [[Pentagon|pentagon]]s
|edges=80
|vertices=40
|symmetry={{math|''D''{{sub|5d}}}}
|vertex_config={{math|20(3.4{{sup|2}}.5)<br>2.10(3.5.3.5)}}
|dual=-
|properties=[[Convex polytope|convex]]
|net=Johnson solid 43 net.png
}}

In [[Geometry|geometry]], the '''elongated pentagonal gyrobirotunda''' or elongated [[Icosidodecahedron|icosidodecahedron]] is one of the [[Johnson solid]]s ({{math|''J''{{sub|43}}}}). As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or [[Icosidodecahedron|icosidodecahedron]] (one of the [[Archimedean solid]]s), by inserting a [[Decagonal prism|decagonal prism]] between its congruent halves. Rotating one of the [[Pentagonal rotunda|pentagonal rotunda]]e ({{math|''J''{{sub|6}}}}) through 36&nbsp;degrees before inserting the prism yields an [[Elongated pentagonal orthobirotunda|elongated pentagonal orthobirotunda]]  ({{math|''J''{{sub|42}}}}).

{{Johnson solid}}

==Formulae==
The following formulae for [[Physics:Volume|volume]] and [[Surface area|surface area]] can be used if all faces are [[Regular polygon|regular]], with edge length ''a'':<ref>[[Biography:Stephen Wolfram|Stephen Wolfram]], "[http://www.wolframalpha.com/input/?i=Elongated+pentagonal+gyrobirotunda Elongated pentagonal gyrobirotunda]" from [[Physics:Wolfram Alpha|Wolfram Alpha]]. Retrieved July 26, 2010.</ref>

:<math>V=\frac{1}{6}\left(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3 \approx 21.5297 a^3</math>
:<math>A=\left(10+\sqrt{30\left(10+3\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2 \approx 39.306 a^2</math>

==References==
{{Reflist}}

==External links==
* {{Mathworld2 | urlname = ElongatedPentagonalGyrobirotunda | title =Elongated pentagonal gyrobirotunda | urlname2 = JohnsonSolid  | title2 = Johnson solid }}

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{{Sourceattribution|Elongated pentagonal gyrobirotunda}}