Moment of Inertia Calculator
The moment of inertia of a solid cylinder, hollow cylinder, sphere, rod or point mass — and the angular momentum and rotational kinetic energy at a given spin.
Moment of inertia is to rotation what mass is to straight-line motion: it says how hard a body is to spin up.
How the moment of inertia calculator works
Moment of inertia is to rotation what mass is to straight-line motion: it says how hard a body is to spin up. It depends on how the mass is distributed — a hollow cylinder has twice the inertia of a solid one of the same mass and radius, because all its mass is at the rim.
Angular momentum is I times angular velocity, and rotational energy is ½ I ω². A flywheel stores energy in exactly this way, which is why flywheels are rims, not discs.
Formula: I = k m r² (k = ½ solid cylinder, 1 hollow, ⅖ sphere, ⅓ rod about its end); L = Iω; E = ½Iω²
Worked examples
| Inputs | Moment of inertia (kg·m²) | Note |
|---|---|---|
| A 12 kg disc at 3,000 rpm | 0.54 | 0.54 kg·m², 26.6 kJ |
| A ring of the same mass | 1.08 | twice the inertia |
| A sphere | 0.432 | two-fifths |
FAQFrequently asked questions
What is moment of inertia?
The rotational equivalent of mass — how much torque it takes to change a body's spin. It depends on mass and, more strongly, on how far that mass sits from the axis.
Why is a ring harder to spin than a disc?
Because all its mass is at the rim, at the full radius. A disc has much of its mass near the center, where it contributes little.
What is angular momentum?
Moment of inertia times angular velocity. It is conserved, which is why a skater spins faster when pulling their arms in.
How much energy can a flywheel store?
Half I ω². Energy goes with the square of speed, so flywheels are spun as fast as their material allows — tens of thousands of rpm for composite ones.
What about a shape not listed?
The parallel-axis theorem and superposition build most shapes from these. A thick-walled cylinder is a solid one minus a smaller solid one.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Institute of Standards and Technology — the US measurement authority
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.