Moment of Inertia Calculator (Common Shapes)
Choose the shape and enter its mass and size.
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Moment of inertia (kg·m²)
Moment of inertia (kg·m²)
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Rotational kinetic energy (J)
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Angular momentum (kg·m²/s)
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Moment of Inertia Calculator (Common Shapes) by TheCalcTools
How the Moment of Inertia Calculator (Common Shapes) works
Moment of inertia measures how hard an object is to spin up. It depends on the mass and how far it sits from the axis: a hoop (all mass at the rim) has I = mr², a solid disc only ½mr².
Rotational energy is ½Iω² and angular momentum Iω, with ω in rad/s.
The formula
Solid sphere ⅖mr², hollow sphere ⅔mr², disc ½mr², hoop mr², rod (centre) mL²/12, rod (end) mL²/3, plate m(a² + b²)/12
Worked example
Using the default values shown in the calculator:
| Shape and axis | Solid sphere (through centre) |
|---|---|
| Mass (kg) | 5 |
| Radius or rod length or plate width (m) | 0.2 |
| Plate height (m) | 0.3 |
| Rotation speed (RPM, for energy) | 600 |
- A 5 kg solid sphere of radius 0.2 m: I = 0.0800 kg·m².
- At 600 RPM it stores 157.91 J.
Result:
| Moment of inertia (kg·m²) | 0.0800 |
|---|---|
| Rotational kinetic energy (J) | 157.91 |
| Angular momentum (kg·m²/s) | 5.0265 |
Frequently asked questions
Why does a solid ball roll down a slope faster than a hoop?
Less of its energy goes into spinning because its moment of inertia is smaller relative to its mass.
Sources and further reading
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This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.