Perpendicular axis theorem

The perpendicular axis 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 formula_1, formula_2, and formula_3 (which meet at origin formula_4) so that the body lies in the formula_5 plane, and the formula_3 axis is perpendicular to the plane of the body. Let "I""x", "I""y" and "I""z" be moments of inertia about axis x, y, z respectively, 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 (or prism, by the stretch rule) has rotational symmetry such that formula_8 and formula_9 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 formula_3 axis is given by:
On the plane, formula_13, so these two terms are the moments of inertia about the formula_1 and formula_2 axes respectively, giving the perpendicular axis theorem.
The converse of this theorem is also derived similarly.
Note that formula_16 because in formula_17, formula_18 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.