Moment of Inertia Calculator
Rotational inertia for the standard shapes, with the parallel axis theorem, radius of gyration and stored energy.
Results
What this tool does
Moment of inertia is to spinning what mass is to moving: the thing that resists being made to go faster. Unlike mass it depends on where the material sits, because every gram counts by the square of its distance from the axis — so a hoop and a solid disc of the same weight behave completely differently. Pick a shape, give the mass and the dimension, and this page gives the inertia about its natural axis, moves the axis if you need it elsewhere, and works out how much energy the object holds at a given speed.
Formula
disc ½mr² · ring mr² · sphere ⅖mr² · shell ⅔mr² · rod about its centre mL²/12 · parallel axes: I = I₀ + m d²
Variables
| Symbol | Meaning | Unit |
|---|---|---|
shape | The shape and the axis | — |
mass | Mass | kg |
radius | Radius | m |
radius_inner | Inner radius | m |
side_a | Length, or the first side | m |
side_b | Side b | m |
offset | How far the real axis is from the centre | m |
rpm | Rotation speed | rpm |
IN | Moment of inertia about the real axis | kg·m² |
I0 | About the shape's own centre | kg·m² |
PA | Added by moving the axis | kg·m² |
GY | Radius of gyration | m |
AV | Angular velocity | rad/s |
RE | Energy stored at that speed | J |
AM | Angular momentum | kg·m²/s |
Worked example
- The shape and the axisdisc
- Mass5 kg
- Radius0.3 m
- Inner radius0 m
- Length, or the first side1 m
- Side b0.5 m
- How far the real axis is from the centre0 m
- Rotation speed0 rpm
- Moment of inertia about the real axis0.22500000 kg·m²
- About the shape's own centre0.22500000 kg·m²
- Added by moving the axis0.00000000 kg·m²
- Radius of gyration0.212132 m
- Angular velocity0.000000 rad/s
- Energy stored at that speed0.000000 J
- Angular momentum0.000000 kg·m²/s
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
Frequently asked questions
Why does a ring have twice the inertia of a disc?
Because inertia counts distance squared, and in a ring every gram sits at the full radius while in a disc most of the material is nearer the middle. Squaring is what makes the difference so large: mass at half the radius contributes a quarter as much. This is also why a flywheel is built with its weight in the rim and why a hollow tube resists twisting better than a solid bar of the same mass — put the material where it counts and you get more inertia for the same weight.
What is the offset for?
For when the object does not spin about its own centre. A door spins about its hinges, not its middle; a pedal about the crank axis, not its own. The parallel axis theorem says you take the inertia about the object's own centre and add the mass times the distance squared. You can see it work above: a rod about its centre is mL²/12, and moving the axis to the end — a distance of L/2 — adds m(L/2)² and gives exactly mL²/3, which is the published value for a rod about its end.