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Moment of Inertia Calculator

Calculate the moment of inertia of common shapes about their principal axis.

About Moment of Inertia Calculator

Moment of inertia depends on where mass sits relative to the axis, not just how much there is. A ring has twice the inertia of a solid disc of the same mass and radius because all its mass is at the rim. Flywheels put mass at the rim to maximise inertia per kilogram, while rotating machinery does the opposite to minimise spin-up torque.

Formula

Moment of Inertia

I = k m r², where k is the shape factor (½ for a disc, 1 for a ring, ⅖ for a sphere)

k
Shape factor determined by how mass is distributed about the axis
radius of gyration
Radius at which a point mass would have the same inertia

Worked example

With the default values below, Moment of Inertia Calculator returns Moment of Inertia of 0.90000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Massminimum 0 kg
20 kg
Radiusminimum 0 m
0.3 m
Shapeone of Solid disc or cylinder, Hollow cylinder or ring, Solid sphere, Hollow sphere, Rod about its centre, Rod about one end
Solid disc or cylinder

Results

Moment of Inertia
0.90000 kg m2
Shape Factor
0.5000
Radius of Gyration
0.2121 m
Energy Stored at 1,000 rpm
4,934.80 J
Torque for 10 rad/s² of Acceleration
9.0000 N m

Step by step

  1. 1Values used
  2. 2Moment of Inertia
  3. 3Moment of Inertia= 0.90000 kg m2
  4. 4Shape Factor= 0.5000
  5. 5Radius of Gyration= 0.2121 m
  6. 6Energy Stored at 1,000 rpm= 4,934.80 J
  7. 7Torque for 10 rad/s² of Acceleration= 9.0000 N m

Assumptions and limits

  • The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 5 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

How is Moment of Inertia calculated?

I = k m r², where k is the shape factor (½ for a disc, 1 for a ring, ⅖ for a sphere) Moment of inertia depends on where mass sits relative to the axis, not just how much there is. A ring has twice the inertia of a solid disc of the same mass and radius because all its mass is at the rim.

Why does Moment of Inertia matter?

Flywheels put mass at the rim to maximise inertia per kilogram, while rotating machinery does the opposite to minimise spin-up torque.

What values do I need to enter?

This calculator takes 3 inputs: Mass, Radius, Shape. The pre-filled defaults are a realistic starting point — replace them with figures from your own environment for a result you can act on.

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