Parallel Axis Theorem Calculator
Enter the moment of inertia about the center of mass, the mass, and the distance to the new axis to get the inertia about any parallel axis — I = I_cm + m·d².
What is the parallel axis theorem?
Shifting the rotation axis
The parallel axis theorem calculator finds the moment of inertia of a body about an axis that runs parallel to one through its center of mass. Steiner's theorem says you take the center-of-mass inertia I_cm and add the mass times the square of the distance between the two axes: I = I_cm + m·d². It turns three measurements — the inertia about the center of mass in kg·m², the mass in kilograms, and the perpendicular distance in metres — into the inertia about the shifted axis. It is the number behind a swinging pendulum bar, a wheel pivoting off-center, and any rigid body rotating about an axis that is not through its own center of mass.
Enter the center-of-mass inertia, the mass, and the distance between the axes to get the moment of inertia about the parallel axis instantly.
The moment of inertia about the parallel axis is the center-of-mass inertia plus the mass multiplied by the square of the distance between the axes.
I = I_cm + m × d²The distance is squared, so a small change in d produces a large change in the added term. Because m·d² can never be negative, the result is always at least I_cm — the center-of-mass axis carries the smallest possible inertia for any given direction. Use kg·m², kilograms, and metres and the answer comes back in kg·m².
Suppose a body has a center-of-mass moment of inertia of 2 kg·m², a mass of 5 kg, and you move the rotation axis 3 m away.
Square the distance
3² = 9 — the squared distance that drives the added term.
Multiply by the mass
5 × 9 = 45 — the mass times the distance squared, in kg·m².
Add the center-of-mass inertia
2 + 45 = 47 kg·m² — the moment of inertia about the parallel axis.
The result tells you how hard it is to angularly accelerate the body about the shifted axis. For the example above, the inertia rose from 2 kg·m² about the center of mass to 47 kg·m² about an axis 3 m away — more than twenty times larger. The crucial insight is that moving the axis away from the center of mass always increases the inertia by exactly m·d², and never decreases it, because mass is positive and the distance is squared. Since the distance enters as a square, doubling d quadruples the added term: at d = 6 m the term grows from 45 to 180 kg·m², four times as large, so the total becomes 182 kg·m². That is why a mass placed far from the pivot is so much harder to spin up than the same mass near it, and why the center-of-mass axis is always the easiest one to rotate about. The center-of-mass inertia and the mass matter too, but the distance is the lever that moves the result the most.
The formula is exact, but a couple of practical points are worth keeping in mind.
Parallel axes and the right distance
The parallel axis theorem only applies when the two axes are genuinely parallel, and d must be the perpendicular distance between them — not the distance to an arbitrary point. The reference axis has to pass through the center of mass; if your known inertia is about some other axis, you cannot add a second m·d² term directly. Keep your units consistent — kg·m² for the inertia, kilograms for the mass, and metres for the distance — or the result will be wrong.