Berechnung des Volumens eines würfels wenn man nur die Raumdiagonale gegeben hat

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 !