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
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[1] 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
are equal,[2] then the perpendicular axes theorem provides the useful relationship: Working in Cartesian coordinates, the moment of inertia of the planar body about the
, so these two terms are the moments of inertia about the
axes respectively, giving the perpendicular axis theorem.
The converse of this theorem is also derived similarly.
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.