Perpendicular-Axis Theorem

The perpendicular axis theorem correlates the moments of inertia of planar objects around axes that are mutually perpendicular and concurrent.

The theorem states that the moment of inertia of a planar object about an axis perpendicular to its plane is equal to the sum of the moments of inertia about two mutually perpendicular axes lying in the plane of the body.

Consider a circular hoop of mass M and radius R lying along an x-y plane. The z-axis is perpendicular to the hoop's plane, and all three axes are concurrent at the hoop's center.

The moment of inertia about the z-axis equals the mass multiplied by the square of its radius.

The circular symmetry of the hoop ensures that the moments of inertia about the planar axes are equal.

Using the perpendicular axis theorem, the moment of inertia along the x-axis equals half the moment of inertia along the z-axis.

As a result, the moment of inertia of the hoop along a planar axis equals half the product of its mass and its radius squared.