Perpendicular axis theorem

The perpendicular axis theorem (or plane figure theorem) states that, "The moment of inertia (I_z) of a laminar body about an axis (z) perpendicular to its plane is the sum of its moments of inertia about two mutually perpendicular axes (x and y) in its plane, all the three axes being concurrent. "

Define perpendicular axes ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ (which meet at origin ${\displaystyle O}$) so that the body lies in the ${\displaystyle xy}$ plane, and the ${\displaystyle z}$ 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. Then the perpendicular axis theorem states that

${\displaystyle I_{z}=I_{x}+I_{y}}$

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 ${\displaystyle I_{x}}$ and ${\displaystyle I_{y}}$ are equal, then the perpendicular axes theorem provides the useful relationship:

${\displaystyle I_{z}=2I_{x}=2I_{y}}$