In physics, the parallel axis theorem or HuygensSteiner 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:
where:
This rule can be applied with the stretch rule and perpendicular axis theorem to find moments of inertia for a variety of shapes.
The parallel axes rule also applies to the second moment of area (area moment of inertia) for a plane region D:
where:
Note: The centroid of D coincides with the center of gravity (CG) of a physical plate with the same shape that has constant density.
In classical mechanics, the Parallel axis theorem (also known as HuygensSteiner theorem) can be generalized to calculate a new inertia tensor J_{ij} from an inertia tensor about a center of mass I_{ij} when the pivot point is a displacement a from the center of mass:
where
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.
