In physics, the parallel axis theorem or Huygens-Steiner theorem can be used to determine the moment of inertia of a rigid body about any axis, given the moment of inertia of the object about the parallel axis through the object's center of mass and the perpendicular distance between the axes.
The moment of inertia about the new axis z is given by:
The parallel axes rule also applies to the second moment of area (area moment of inertia) for a plane region D:
In classical mechanics, the Parallel axis theorem (also known as Huygens-Steiner theorem) can be generalized to calculate a new inertia tensor Jij from an inertia tensor about a center of mass Iij when the pivot point is a displacement a from the center of mass:
is the displacement vector from the center of mass to the new axis, and
is the Kronecker delta.
We can see that, for diagonal elements (when i = j), displacements perpendicular to the axis of rotation results in the above simplified version of the parallel axis theorem.