Perpendicular axis theorem
The perpendicular axis theorem (or plane figure theorem) states that the moment of inertia of a planar lamina (i.e. 2-D body) about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia of the lamina about the two axes at right angles to each other, in its own plane intersecting each other at the point where the perpendicular axis passes through it. Define perpendicular axes , , and  (which meet at origin ) so that the body lies in the  plane, and the  axis is perpendicular to the plane of the body. Let Ix, Iy and Iz be moments of inertia about axis x, y, z respectively. Then the perpendicular axis theorem states that

This rule can be applied with the parallel axis theorem and the stretch rule to find polar moments of inertia for a variety of shapes. If a planar object has rotational symmetry such that  and  are equal,
then the perpendicular axes theorem provides the useful relationship:

Derivation 
Working in Cartesian coordinates, the moment of inertia of the planar body about the  axis is given by:

On the plane, , so these two terms are the moments of inertia about the  and  axes respectively, giving the perpendicular axis theorem. The converse of this theorem is also derived similarly. Note that  because in ,  measures the distance from the axis of rotation, so for a y-axis rotation, deviation distance from the axis of rotation of a point is equal to its x coordinate. References

See also
 Parallel axis theorem
 Stretch rule

Rigid bodies
Physics theorems
Articles containing proofs
Classical mechanics
Moment (physics)