Inscribed sphere of rhombic triacodahedron

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.

Solution