Perpendicular axis theorem

The perpendicular axis theorem (or plane figure theorem) states that, "The moment of inertia (Iz) 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 $$x$$, $$y$$, and $$z$$ (which meet at origin $$O$$) so that the body lies in the $$xy$$ plane, and the $$z$$ 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


 * $$I_z = I_x + I_y$$

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 $$I_x$$ and $$I_y$$ are equal, then the perpendicular axes theorem provides the useful relationship:


 * $$I_z = 2I_x = 2I_y$$

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


 * $$I_{z} = \int (x^2 + y^2) \,dm = \int x^2\,dm + \int y^2\,dm = I_{y} + I_{x}$$

On the plane, $$z=0$$, so these two terms are the moments of inertia about the $$x$$ and $$y$$ axes respectively, giving the perpendicular axis theorem. The converse of this theorem is also derived similarly.

Note that $$\int x^2\,dm = I_{y} \ne I_{x}$$ because in $$\int r^2\,dm $$, $$r$$ 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.