A thin, rectangular sheet of metal has mass M and sides of length a and b. Find the moment of inertia of this sheet about an axis that lies in the plane of the plate, passes through the center of the plate, and is parallel to the side with length b.
The moment of inertia about an axis through the centre and perpendicular to the lamina is: M(a^2 + b^2) / 12.
If h is the length of a semi-diagonal: h^2 = (a/2)^2 + (b/2)^2 = (a^2 + b^2) / 4.
By the parallel axis theorem, the moment of inertial about an axis through the corner is: M(a^2 + b^2) / 12 + Mh^2 = M(a^2 + b^2)(1/12 + 1/4) = M(a^2 + b^2) / 3.
