Vì $\left( A'B'C'D' \right)//\left( ABCD \right)\Rightarrow A'B'//AB,B'C'//BC,C'D'//CD$

Mà: $\frac{SA'}{SA}=\frac{1}{3}\Rightarrow \frac{SB'}{SB}=\frac{SC'}{SC}=\frac{SD'}{SD}=\frac{1}{3}$. Gọi ${{V}_{1}},{{V}_{2}}$ lần lượt là ${{V}_{S.ABC}},{{V}_{S.ACD}}$

Ta có: ${{V}_{1}}+{{V}_{2}}=V$

$\frac{{{V}_{S.A'B'C'}}}{{{V}_{S.ABC}}}=\frac{SA'}{SA}.\frac{SB'}{SB}.\frac{SC'}{SC}=\frac{1}{27}\Leftrightarrow {{V}_{S.A'B'C'}}=\frac{{{V}_{1}}}{27}$.

$\frac{{{V}_{S.A'C'D'}}}{{{V}_{S.ACD}}}=\frac{SA'}{SA}.\frac{SC'}{SC}.\frac{SD'}{SD}=\frac{1}{27}\Leftrightarrow {{V}_{S.A'C'D'}}=\frac{{{V}_{2}}}{27}$.

Vậy ${{V}_{S.A'BC'D'}}={{V}_{S.A'B'C'}}+{{V}_{S.A'C'D'}}=\frac{{{V}_{1}}+{{V}_{2}}}{27}=\frac{V}{27}$.

Vậy ${{V}_{S.A'BC'D'}}=\frac{V}{27}$.