The moment of inertia for an area that lies in the x-y plane can also be calculated about the z axis, which is known as the polar moment of inertia. The polar moment of inertia of the area A is calculated as

If the polar moment of inertia is calculated at the centroid of the area, it is denoted

J_{x'} = I_{x'} + I_{y'}

The polar moment of inertia is commonly used when calculating the torsion of shafts.

Just as the centroid of an area can be calculated by breaking the area into simpler composite parts, the moments of inertia of a complicated area can be calculated by breaking the area into simpler composite parts.

For an arbitrary axis, the moments of inertia for an area made of composite parts are given by

This technique also works for polar moment of inertia.

Subtraction of Material (Holes)

Previously, in the
Centroid: Composite Parts section, it was shown that when the centroid of line, area, or volume are calculated, holes or cutouts can be accounted for by considering them a negative line length, area, or volume respectively. When calculating moments of inertia, we can deal with cutouts and holes in the same manner. Both the moments of inertia about the centroid of the hole as well as the area of the hole are considered negative when we are summing the composite parts.