Perpendicular Axis Theorem

In general, for any 2D distribution of mass, the moment of inertia about an axis orthogonal to the plane of the mass equals the sum of the moments of inertia about any two mutually orthogonal axes in the plane of the mass intersecting the first axis. To see this, consider an arbitrary mass element $dm$ having rectilinear coordinates $(x_1,x_2,0)$ in the plane of the mass. (All three coordinate axes intersect at a point in the mass-distribution plane.) Then its moment of inertia about the axis orthogonal to the mass plane is $dm\,(x_1^2+x_2^2)$ while its moment of inertia about coordinate axes within the mass-plane are respectively $dm\,x_1^2$ and $dm\,x_2^2$ . This, the perpendicular axis theorem is an immediate consequence of the Pythagorean theorem for right triangles.