Guten Abend Anonymous,
Raumdiagonale $${{\mathtt{e}}}^{{\mathtt{2}}} = {{\mathtt{d}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{{\mathtt{a}}}^{{\mathtt{2}}}$$ ; $${{\mathtt{d}}}^{{\mathtt{2}}} = {{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{{\mathtt{a}}}^{{\mathtt{2}}}$$ ; $${{\mathtt{e}}}^{{\mathtt{2}}} = {{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{{\mathtt{a}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{{\mathtt{a}}}^{{\mathtt{2}}} = {\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{a}}}^{{\mathtt{2}}}$$
Würfel-Raumdiagonale $${\mathtt{e}} = {\mathtt{a}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}$$ => $${\mathtt{a}} = {\frac{{\mathtt{e}}}{{\sqrt{{\mathtt{3}}}}}}$$
Volumen $${\mathtt{V}} = {{\mathtt{a}}}^{{\mathtt{3}}}$$ = $${\left({\frac{{\mathtt{e}}}{{\sqrt{{\mathtt{3}}}}}}\right)}^{{\mathtt{3}}}$$ = $${\frac{{{\mathtt{e}}}^{{\mathtt{3}}}}{\left({\mathtt{3}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}\right)}}$$
Kontrollrechnung: e = 10 cm V = 192,45 cm³
$${\mathtt{V}} = {\frac{{{\mathtt{10}}}^{{\mathtt{3}}}}{\left({\mathtt{3}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}\right)}} \Rightarrow {\mathtt{V}} = {\mathtt{192.450\: \!089\: \!729\: \!875\: \!254\: \!8}}$$
Gruß radix 
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