Theorem of the perpendicular axis states that if IXX and IYY 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 IZZ about the axis Z-Z, perpendicular to the plane and passing through the intersection of X-X and Y-Y is given by
IZZ = IXX + IYY.
The moment of intertia IZZ 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.