Theorem of the perpendicular axis states that if I_XX and I_YY be the moment of intertia of a plane section about two mutually perpendicular axis X-X and Y-Y in the plane of the section, then the moment of intertia of the section I_ZZ about the axis Z-Z, perpendicular to the plane and passing through the intersection of X-X and Y-Y is given by

I_ZZ = I_XX + I_YY.

The moment of intertia I_ZZ is also known as polar moment of intertia.

Proof. A plane section of area A and lying in plane x-y is shown in Fig. 4.20. Let OX and OY be the two mutually perpendicular axes, and OZ be the perpendicular axis. Consider a small area dA.

Let x = Distance of dA from the axis OY

y = Distance of dA from axis OX

r = Distance of dA from axis OZ

The above equation shows that the moment of intertia of an area about an axis at origin normal to x, y plane is the sum of moments of intertia about the corresponding x and y-axis.