Parallel Perpendicular Axis Theorem

A theorem which states that the moment of inertia of a body about any given axis is the moment of inertia about a parallel axis through the center of mass, plus the moment of inertia that the body would have about the given axis if all the mass of the body were located at the center of mass. The moment of inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of the moments of inertia of the lamina about any two mutually perpendicular axes, passing in its own plane, intersecting each other at the point through which the perpendicular axis passes

For a planar object, the moment of inertia about an axis perpendicular to the plane is the sum of the moments of inertia of two perpendicular axes through the same point in the plane of the object. The utility of this theorem goes beyond that of calculating moments of strictly planar objects. It is a valuable tool in the building up of the moments of inertia of three dimensional objects such as cylinders by breaking them up into planar disks and summing the moments of inertia of the composite disks.

The parallel-axis theorem is a theorem of physics that makes it easier to find some moments of inertia. You have a rigid object (mass is m) whose axis of rotation passes through its center of mass. Now you have another axis, parallel to this axis, and you want to know the moment of inertia of the object about this particular axis. Let's call distance between these two axes is d. The moment of inertia I of the object through an axis parallel to that axis that passes through its center of mass is:

I = I_CM + m d²

Where I_CM is the moment of inertia about an axis that passes through the center of mass. N.B., this axis must be PARALLEL.