Đặt $SH=x$ .suy ra  $V=\frac{1}{3}x.\left( \frac{1}{2}a.a\sqrt{3} \right)=\frac{{{a}^{3}}\sqrt{6}}{6}$

                                 $\Leftrightarrow x=\frac{{{a}^{3}}\sqrt{6}}{6}.\frac{6}{{{a}^{2}}\sqrt{3}}=a\sqrt{2}$

Ta có  $d\left( C,\left( SAB \right) \right)=2d\left( H,\left( SAB \right) \right)=2HK$ 

mà   $\frac{1}{H{{K}^{2}}}=\frac{1}{2{{a}^{2}}}+\frac{4}{3{{a}^{2}}}\Rightarrow HK=\frac{a\sqrt{66}}{11}$

          $d\left( C,\left( SAB \right) \right)=\frac{2a\sqrt{66}}{11}.$