$\begin{align}  & SH=\sqrt{S{{A}^{2}}-A{{H}^{2}}=2a} \\  & \Rightarrow AM=AC.\sin \angle MCA=AC.\sin \angle ASH=AC.\frac{AH}{SA}=\frac{4a\sqrt{5}}{5} \\ \end{align}$

$\begin{align}  & \Rightarrow \frac{AM}{AS}=\frac{4}{5}\Rightarrow {{S}_{SMC}}=\frac{{{S}_{SAC}}}{5} \\  & \Rightarrow {{V}_{B.SMC}}=\frac{{{V}_{B.SAC}}}{5}=\frac{{{V}_{S.ABCD}}}{10}=\frac{SH.AC.BD}{60}=\frac{4{{a}^{3}}}{15} \\ \end{align}$