Theorem of Perpendicular Axis

According to this theorem, the moment of inertia of a plane lamina (i.e. a two dimensional body of any shape or size) about any axis OZ perpendicular to the plane of the lamina is equal to sum of the moments of inertia of the lamina about any two mutually perpendicular axes OX and OY in the plane of the lamina, meeting at a point where the given axis OZ passes through the lamina.

In the figure below,

$\displaystyle {{I}_{x}}$= moment of inertia of the lamina about OX

$\displaystyle {{I}_{y}}$= moment of inertia of the lamina about OY

$\displaystyle {{I}_{z}}$= moment of inertia of the lamina about OZ

According to the theorem of perpendicular axes,

$\bbox[15px, #e4e4e4, border: 2px solid #000000]{\boldsymbol {\displaystyle {{I}_{z}}={{I}_{x}}+{{I}_{y}}}}$