How to find that Dehn invariant of a dodecahedron?

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What is the Dehn invariant of a regular dodecahedron with center (0,0,0), and radius 1?

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$$ \begin{align} D &= \sum_{e} \ell(e) \otimes \theta(e) \\ &= 30 \cdot \frac{4}{\sqrt{3} \left( 1 + \sqrt{5} \right)} \otimes \arccos \left( -\frac{1}{\sqrt{5}} \right) \\ &\equiv 10\sqrt{3} \left( \sqrt{5} - 1 \right) \otimes \arccos \left( -\frac{1}{\sqrt{5}} \right) \in \Bbb{R} \otimes_{\Bbb{Q}} \big(\Bbb{R}/(\Bbb{Q}\pi) \big)\end{align} $$