Theorem of parallel & perpendicular axes

Theorem of Parallel Axes:

In order to apply this theorem to any object, we need two axes parallel to each other with one of them passing through the centre of mass of the object. 

Figure shows an object of mass M. Axis MOP is any axis passing through point O. Axis ACB is passing through the centre of mass C of the object, parallel to the axis MOP, and at a distance h from it (h = CO). Consider a mass element dm located at point D. Perpendicular on OC (produced) from point 

It states that, “The moment of inertia (IO) of an object about any axis is the sum of its moment of inertia (IC) about an axis parallel to the given axis, and passing through the centre of mass and the product of the mass of the object and the square of the distance between the two axes (Mh2).” 

Theorem of Perpendicular Axes:

This theorem relates the moment of inertias of a laminar object about three mutually perpendicular and concurrent axes, two of them in the plane of the object and the third perpendicular to the object. Laminar object is like a leaf, or any two dimensional object, e.g., a ring, a disc, any plane sheet, etc. 

Figure shows a rigid laminar object able to rotate about three mutually perpendicular axes x, y and z. Axes x and y are in the plane of the object while the z axis is perpendicular to it, and all are concurrent at O. Consider a mass element dm located at any point P. PM = y and PN = x are the perpendiculars drown from P respectively on the x and y axes. The 

It states that, “The moment of inertia (IZ ) of a laminar object about an axis (z) perpendicular to its plane is the sum of its moments of inertia about two mutually perpendicular axes (x and y) in its plane, all the three axes being concurrent”.