The moment of inertia of semicircular plate of radius R and mass M about axis AA' in its plane passing through its centre is

Moment of inertia of disk w.r.t. principle axis (1) =MR22Moment of inertia of disk w.r.t. diametrical axis (2)From perpendicular axes theoremIz=Ix+Iy…(i)here Ix=Iy, because both are diametrical axis,So here  Iz=MR22and Ix=Iy=I( Let )So from eq. (i)⇒MR22=I+I⇒I=MR24For half disk also, it is applicable, SoI=MR24(It does not depend on θ given in diagram, because given axis is also similar to diametrical axis)