Chọn D
Đặt: \(x=\frac{SA}{SA} =1;\, y=\frac{SM}{SB} =\frac{1}{2} ;\, z=\frac{SI}{SC} ;\, t=\frac{SN}{SD} =\frac{1}{2} .\)
Ta có: \(\frac{SA}{SA} +\frac{SC}{SI} =\frac{SB}{SM} +\frac{SD}{SN} \Leftrightarrow 1+\frac{SC}{SI} =2+2\Leftrightarrow \frac{SC}{SI} =3\Rightarrow z=\frac{1}{3} . \)
Mà: \(\frac{V_{S.AMIN} }{V_{S.ABCD} } =\frac{xyzt}{4} \left(\frac{1}{x} +\frac{1}{y} +\frac{1}{z} +\frac{1}{t} \right)=\frac{1+2+3+2}{4.1.2.3.2} =\frac{1}{6} \Rightarrow V_{S.AMIN} =\frac{1}{6} V_{S.ABCD} .\)
\(\[\Rightarrow V_{ABCDMNI} =\frac{5}{6} V_{S.ABCD} =\frac{5}{6} .\frac{1}{3} .a\sqrt{3} .a^{2} =\frac{5\sqrt{3} }{18} a^{3} .\] \)