Four point masses, each of value $m$, are placed at the corners of a square $ABCD$ of side $\ell$. The moment of inertia through $A$ and parallel to $BD$ is:

The diagonal $BD$ has length $\ell \sqrt{2}$. The axis passes through $A$ and is parallel to $BD$.Perpendicular distances of the masses from the axis:Mass at $A$: $r_A=0$ Mass at $B$: $r_B=\text{projection of AB on perpendicular to axis}=\frac{\ell }{\sqrt{2}}$ Mass at $D$: $r_D=\frac{\ell }{\sqrt{2}}$ Mass at $C$: Distance from $A$ along diagonal $AC=\ell \sqrt{2}$.Total $I=m(0{)}^2+m(\ell /\sqrt{2}{)}^2+m(\ell /\sqrt{2}{)}^2+m(\ell \sqrt{2}{)}^2$$I=0+m{\ell }^2/2+m{\ell }^2/2+2m{\ell }^2$$I=m{\ell }^2+2m{\ell }^2=3m{\ell }^2$.