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## Inscribed sphere of rhombic triacontahedron

Projective Geometry, Solid Geometry
Grigorios Kostakos
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### Inscribed sphere of rhombic triacontahedron

Consider a rhombic triacontahedron $R$ with edge length $1$ and the inscribed sphere $S$ of $R$ (tangent to each of the rhombic triacontahedron's faces). Prove that the radius $r$ of $S$ has length $r=\frac{\Phi^2}{\sqrt{1 + \Phi^2}} =\frac{3 + \sqrt{5}}{\sqrt{10 + 2\sqrt{5}}}\,,$ where $\Phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio.
Grigorios Kostakos
Grigorios Kostakos
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Joined: Mon Nov 09, 2015 1:36 am
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### Re: Inscribed sphere of rhombic triacontahedron

A rhombic triacontahedron has $30$ faces, all of which are golden rhombi. A golden rhombus is a rhombus such that the ratio of the long diagonal $\varDelta$ to the short diagonal $\delta$ is equal to the golden ratio $\Phi$, ie
$\frac{\varDelta}{\delta}=\Phi=\frac{1+\sqrt{5}}{2}\quad(1)\,.$Also, the short diagonals $\delta$s are the edges of a dodecahedron $T$, which is inscribed in the rhombic triacontahedron $R$. ( In the first figure, the polygon $ABCDEA$ is a face of the dodecahedron. )
inscribed_sphere_r30edron_i.png (73.58 KiB) Viewed 2028 times
[/centre]

The sphere $S$ which is inscribed in $R$ (tangent to each of the rhombic triacontahedron's faces) touches the center of each golden rhombus, ie the point of intersection of its diagonals. Also, the same sphere $S$ is circumscribed in dodecahedron $T$ ( passes from dodecahedron's vertices ).
So, if $O$ is the center of the sphere $S$ and $H$ is the center of the golden rhombus $AFBG$, we have that $r=OH$. (2nd figure)
inscribed_sphere_r30edron_ii.png (73.84 KiB) Viewed 2027 times
[/centre]

From the properties of dodecahedron, it is known that $r=\frac{\Phi^2}{2}\,AB=\frac{\Phi^2}{2}\,\delta\quad(2)\,.$ From the Pythagorean theorem in $\triangle{AGH}$ we have that \begin{align*}
Finally, from $(2)$ we have that
$r=\frac{\Phi^2}{2}\,\delta=\frac{\Phi^2}{2}\,\frac{2}{\sqrt{1+\Phi^2}}=\frac{\Phi^2}{\sqrt{1 + \Phi^2}} =\frac{3 + \sqrt{5}}{\sqrt{10 + 2\sqrt{5}}}\,.$