Perpendicular axis theorem: It states “MI of a plane lamina about an axis perpendicular to the plane of lamina and passing through the centroid of the lamina is equal to the addition of the moments of inertia of the lamina about its centroidal axes”.

Figure below shows the plane lamina laying in XY plane, OX and OY are mutually perpendicular and OZ is the axis perpendicular to plane XY of the lamina.

MI of lamina about OZ is

$\mathrm{I}_{\mathrm{z}}=\Sigma d A\left(r^{2}\right)$

$\mathrm{I}_{\mathrm{z}}=\Sigma d A\left(x^{2}+y^{2}\right)$

$\mathrm{I}_{\mathrm{z}}=\Sigma d A\left(x^{2}\right)+\Sigma d A\left(y^{2}\right)$

$I_{z}=I_{x}+I_{y}$