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.

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