Last updated: 2026-06-15

Moment of Inertia

Moment of inertia is to rotation what mass is to straight-line motion: it tells you how stubbornly an object resists a change in its rate of spin. But unlike mass, it depends on where the material sits relative to the axis — mass far from the axis matters far more than mass close to it. This calculator gives the moment of inertia of the standard textbook shapes from their mass and size.

The general form

For a body of mass $m$ whose characteristic size is $r$, the standard shapes all share the form

where $c$ is a dimensionless coefficient set by the shape and the axis. The radius appears squared, which is why a wheel with a heavy rim is so much harder to spin up than a solid disk of the same mass: moving mass outward raises $I$ steeply.

Coefficients for common shapes

For the rod, $L$ is the full length; for every other shape, $r$ is the radius.

Worked example

A solid disk of mass $m = 2\ \text{kg}$ and radius $r = 0.5\ \text{m}$, spinning about its central axis:

Swap the disk for a hoop of the same mass and radius and the coefficient jumps to $c_\text{hoop} = 1$, doubling the moment of inertia to $0.5\ \text{kg·m}^2$ — because all the hoop's mass sits out at the rim.

Shifting the axis

The coefficients above assume specific axes. To move the axis somewhere parallel, use the parallel-axis theorem,

where $I_\text{cm}$ is the moment of inertia about a parallel axis through the centre of mass and $d$ is the gap between the axes. The "rod about one end" entry is just this theorem applied to a rod with $d = \tfrac{L}{2}$, which lifts $\tfrac{1}{12}$ up to $\tfrac{1}{3}$.

Why it matters

Once you have $I$, the rest of rotational mechanics follows: the rotational kinetic energy is $\tfrac{1}{2}I\omega^2$, the angular momentum is $I\omega$, and the torque needed for a given angular acceleration is $\tau = I\alpha$. Getting the moment of inertia right is the first step in all of these.