parallel-axis theorem
For any axis parallel to one through the centre of mass, I = I_CM + M·d² (also called Steiner's theorem).
Definition
The parallel-axis theorem relates the moment of inertia of a rigid body about any axis to the moment of inertia about the parallel axis through the body's centre of mass. If I_CM is the centre-of-mass value and d is the perpendicular distance between the two axes, then the moment of inertia about the offset axis is I = I_CM + M·d², where M is the total mass of the body.
The theorem is a direct consequence of the definition of the center of mass and can be proved in two lines by expanding ∫(r − d)² dm. The middle cross-term vanishes because the first moment of the mass distribution about the centre of mass is zero by definition.
It is one of the most practically useful results in rigid-body mechanics. A thin rod of length L has I_CM = M·L²/12 about its centre; by the theorem, its moment about one end (d = L/2) is M·L²/12 + M·(L/2)² = M·L²/3 — a result that would otherwise require recomputing the integral from scratch. Together with the perpendicular-axis theorem (for laminar bodies), it turns a short list of fundamental integrals into a large table of moments of inertia about any reasonable axis. The theorem is often called Steiner's theorem after the Swiss geometer who published it in the mid-nineteenth century.