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Truncated Dodecahedron

\begin{figure}\begin{center}\BoxedEPSF{Truncated_Dodecahedron_net.epsf scaled 800}\end{center}\end{figure}

An Archimedean Solid whose Dual Polyhedron is the Triakis Icosahedron. It has Schläfli Symbol t$\{5,3\}$. It is also Uniform Polyhedron $U_{26}$ and has Wythoff Symbol $2\,3\,\vert\,5$. Its faces are $20\{3\}+12\{10\}$. The Inradius, Midradius, and Circumradius for $a=1$ are

$\displaystyle r$ $\textstyle =$ $\displaystyle {\textstyle{5\over 488}}(17\sqrt{2}+3\sqrt{10}\,)\sqrt{37+15\sqrt{5}} \approx 2.88526$  
$\displaystyle \rho$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}(5+3\sqrt{5}\,) \approx 2.92705$  
$\displaystyle R$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}\sqrt{74+30\sqrt{5}} \approx 2.96945.$  

© 1996-9 Eric W. Weisstein