Parallel perpendicular axis theorem

This theorem states that the moment of inertia of the body about any given axis is the moment of inertia about the parallel axis through the center of mass, plus the moment of inertia which 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 2 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 2 perpendicular axes through the same point in the plane of the object. The applications of this theorem go beyond that of calculating moments of planar objects. It is a important tool in the building up of the moments of inertia of 3 dimensional objects like cylinders by breaking them up into the planar disks and summing the moments of inertia of composite disks.

The parallel-axis theorem is a theorem of physics which makes it simple to find some moments of inertia.  You have a rigid object whose axis of rotation passes through the 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 axis. Let's call distance between these 2 axes is d. The moment of inertia I of the object through an axis parallel to that axis which passes through its center of mass can be given as:

I = I_CM + m d²

Here I_CM is moment of inertia about the axis which passes through the center of mass. N.B., this axis should be PARALLEL.

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