I recently got to learn about rotational mechanics and learnt a bit about the parallel axis theorem.
It states that the moment of inertia of a rigid body about an axis YY is equal to the moment of inertia about another axis Y’Y’ passing through the center-of-mass G of the body in a direction parallel to YY, plus the product of total mass $M$ of the body and square of the perpendicular distance between the two parallel axes.
However, I am unable to understand the technique used in the solution to this question:
A symmetrical square lamina of mass $M$ has uniform semi-circular plates attached to it on its four sides as shown in the figure. Each plate has the same mass $M$, and the disc of a square is equal to $a$. Calculate the moment of inertia of system about an axis passing through the center $O$, perpendicular to the plane of the lamina.
I think that the solution calculates the moment of inertia of the semi circular mass along the center of square. According to the parallel axis theorem, the moment of inertia along the center of square can be calculated if we know the moment of inertia along the center of mass of the semicircular plate, which obviously doesn't lie on the common periphery of the square and the plates. Am I right or am I missing something basic?