Moment of Inertia Calculator

Moment of inertia is the rotational counterpart of mass: it measures how much an object resists a change in its spin. It grows with both the total mass and, more sharply, with how far that mass sits from the rotation axis, since each bit of mass contributes its mass times the square of its distance.

Six standard cases: a solid cylinder or disk (I = 1/2 m r^2), a thin hoop or ring (m r^2), a solid sphere (2/5 m r^2), a hollow spherical shell (2/3 m r^2), a thin rod about its centre (1/12 m L^2), and a thin rod about one end (1/3 m L^2). Each uses the common symmetry axis for that body.

All of a hoop's mass sits out at the rim, the farthest point from the axis, so it uses the full I = m r^2. A disk spreads its mass from the centre outward, so the average squared distance is smaller and it gives I = 1/2 m r^2, exactly half the hoop for the same mass and radius.

About the centre the mass is split evenly on both sides and stays relatively close, giving 1/12 m L^2. Pivot at one end and much more of the rod lies far from the axis, giving 1/3 m L^2, which is exactly four times larger for the same rod.

Enter mass in kilograms and the radius or length in metres, the SI base units. The answer then comes out in kilogram metre squared, written kg x m^2. If you have grams or centimetres, convert them first, since the calculator does not rescale units.

Yes. The rod formulas assume a thin rod whose thickness is negligible next to its length, and the hoop assumes all mass at a single radius. For a thick disk, a ring with real width, or a rod with a large cross-section the true value is slightly different, though usually close.

Not directly. These are the standard central or end axes only. To move to a parallel axis a distance d away, apply the parallel-axis theorem yourself: I_new = I + m d^2, using the value this tool gives as the starting I.

Mass, radius and length must each be greater than zero, so a zero or negative entry returns a short error instead of a result. Non-numeric text is also rejected. Fix the value and the result reappears instantly.

The value plugs straight into rotational equations: rotational kinetic energy is 1/2 I omega^2, angular momentum is I omega, and torque equals I times angular acceleration. Keeping I in kg x m^2 and angles in radians keeps those formulas in consistent SI units.