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Moment of Inertia of a Disk Calculator

Enter the disk's mass and radius.

How it is calculated

For a uniform solid disk spinning about the axis through its centre and perpendicular to its face, the tool uses I = ½ × m × r². m is the mass in kilograms and r is the radius in metres, so the result is in kg·m². It is shown to four decimals, and the tool also echoes the formula with your numbers filled in.

Both values must be greater than zero. The radius is squared, so doubling it makes I four times larger, while doubling the mass only doubles I. The formula assumes uniform density and rotation about the central axis; a disk with a hole, a ring, or rotation about its edge or a diameter needs a different formula. Because of the four-decimal rounding, very light, small parts can show 0.0000 even though their inertia is not exactly zero.

Worked example

A uniform steel disk of mass 12 kg and radius 0.25 m. Radius squared = 0.25 × 0.25 = 0.0625 m². Then I = ½ × 12 × 0.0625 = 6 × 0.0625 = 0.3750 kg·m². If the same disk had a 0.5 m radius, r² would be 0.25 and I would be 1.5 kg·m², four times as much, although the mass did not change.

Try this example →

Common mistakes

  • Entering the diameter instead of the radius. Since r is squared, this makes the result four times too large.
  • Entering the radius in centimetres or millimetres. The tool works in metres, so 25 cm must be typed as 0.25.
  • Entering the mass in grams or in pounds. The mass is in kilograms.
  • Using this formula for a hollow ring, a disk with a large hole or a rotation axis through the rim. Those cases have different inertia values.
  • Reading 0.0000 as zero for a tiny part. The result is rounded to four decimals, so compute small parts in larger units or by hand.

About this calculator

Moment of inertia measures how much an object resists changes to its rotational motion, playing the same role in rotational dynamics that mass plays in linear motion. For a uniform solid disk spinning about the axis through its center, perpendicular to its face, the moment of inertia is given by I = ½·m·r², where mass concentrated closer to the axis contributes less resistance to rotation than mass farther out — which is why a disk's moment of inertia is exactly half that of a ring or hoop with the same mass and radius.

This formula appears constantly in mechanical and automotive engineering — for flywheels, brake rotors, gears, and pulleys — as well as in introductory physics courses covering rotational kinetic energy and angular momentum. Engineers use it to size flywheels for energy storage, calculate the torque needed to spin up a rotating component, and predict how quickly a disk-shaped part will accelerate under a given applied torque.

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Comments (1)

  • Editorial team

    The inertia is mostly a stepping stone. Once you have it, the torque needed to change the spin is the inertia times the angular acceleration you want: with the default disk of 0.2250 kg·m², every 1 rad/s² of acceleration costs 0.225 N·m, ignoring friction. For a real part, weigh it and measure the radius rather than trusting a catalogue figure. What are you spinning up: a flywheel, a rotor, a turntable?

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