We can easily find R using the Pythagoras theorem,
⇒R=√ a2−(a/2)2 =a√2
We know that for a square plate, the moment of inertia along a perpendicular axis passing through the centre of mass is,
⇒ Iperpendicular = ma2/6
So, using the parallel axis theorem, we get
⇒Iparallel = Iperpendicular + MR2
Here Iparallel is the moment of inertia along the parallel axis, Iperpendicular is the moment of inertia along the axis through the centre of mass, M is the mass of the object and R is the distance between the centre of mass and the parallel axis.
Substituting the value of R and Iperpendicular we get ⇒ Iparallel = Iperpendicular + MR2
∴Iparallel =ma2/6 + ma2/(√2)2 = 2/3ma2