Perpendicular axes rule

In physics, the perpendicular axis theorem (or plane figure theorem) can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis perpendicular to the plane, given the moments of inertia of the object about two perpendicular axes lying within the plane. The axes must all pass through a single point in the plane.

Define perpendicular axes ${\displaystyle x\,}$, ${\displaystyle y\,}$, and ${\displaystyle z\,}$ (which meet at origin ${\displaystyle O\,}$) so that the body lies in the ${\displaystyle xy\,}$ plane, and the ${\displaystyle z\,}$ axis is perpendicular to the plane of the body. Let I_x, I_y and I_z be moments of inertia about axis x, y, z respectively, the perpendicular axis theorem states that

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