Perpendicular axis theorem

The perpendicular axis theorem (or plane figure theorem) states that for a planar lamina the moment of inertia about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia about two mutually perpendicular axes in the plane of the lamina, which intersect at the point where the perpendicular axis passes through. This theorem applies only to planar bodies and is valid when the body lies entirely in a single plane. Define perpendicular axes x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} (which meet at origin O {\displaystyle O} ) so that the body lies in the x y {\displaystyle xy} plane, and the z {\displaystyle z} 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 I z = I x + I y {\displaystyle I_{z}=I_{x}+I_{y}} 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 I x {\displaystyle I_{x}} and I y {\displaystyle I_{y}} are equal, then the perpendicular axes theorem provides the useful relationship: I z = 2 I x = 2 I y {\displaystyle I_{z}=2I_{x}=2I_{y}}