This theorem is applicable only to the plane laminar bodies
This theorem states that, the moment of inertia of a plane laminar about an axis perpendicular to its plane is equal to the sum of the moment of inertia of the lamina about two axis mutually perpendicular to each other in its plane and intersecting each other at the point where perpendicular axis passes through it
Consider plane laminar body of arbitrary shape lying in the x-y plane as shown below in the figure
The moment of inertia about the z-axis equals to the sum of the moments of inertia about the x-axis and y axis
To prove it consider the moment of inertia about x-axis
where sum is taken over all the element of the mass m_{i}
The moment of inertia about the y axis is
Moment of inertia about z axis is
where r_{i} is perpendicular distance of particle at point P from the OZ axis
For each element
r_{i}^{2}=x_{i}^{2} + y_{i}^{2}
Watch this tutorial for more information on Moment of Inertia