Moment of Inertia Calculator
Calculate the moment of inertia for common shapes: solid disk, hollow cylinder, solid sphere, hollow sphere, thin rod, and rectangular plate.
The moment of inertia (I) is the rotational equivalent of mass. It measures an object’s resistance to angular acceleration:
τ = Iα (analog of F = ma for rotation)
The moment of inertia depends on both the mass and how that mass is distributed around the rotation axis. Mass farther from the axis contributes more.
Formulas by shape:
| Shape | Formula | About which axis |
|---|---|---|
| Solid disk/cylinder | I = ½MR² | Central axis |
| Hollow cylinder | I = ½M(R₁² + R₂²) | Central axis |
| Solid sphere | I = (2/5)MR² | Any diameter |
| Hollow sphere (thin shell) | I = (2/3)MR² | Any diameter |
| Thin rod | I = (1/12)ML² | Through center |
| Thin rod | I = (1/3)ML² | Through one end |
| Rectangular plate | I = (1/12)M(a² + b²) | Through center |
Parallel axis theorem: To find I about any axis parallel to the center-of-mass axis: I = I_cm + Md² where d is the perpendicular distance between the axes.
Physical intuition: A solid sphere has less MOI than a hollow sphere of the same mass and radius, because more mass sits near the center. This is why bowling balls can roll faster than hollow balls of the same weight: they have lower rotational inertia.
The coefficient is the whole story
Every formula in the table has the same shape: some fraction times M times a length squared. Only the fraction changes, and it runs from 1/12 for a rod spun about its middle up to 2/3 for a hollow sphere. That range of eight is the entire effect of geometry. Double the radius instead and you quadruple the inertia, because the length term is squared. Distance from the axis matters far more than how the mass is arranged at a given distance.
That squared term is why a figure skater speeds up by pulling their arms in, and why a flywheel is built as a heavy rim on a light spoke rather than a solid slab. Putting the metal where the radius is largest buys the most stored energy per kilogram.
Rods have two entries for a reason
The same rod has four times the moment of inertia about its end as about its middle, 1/3 versus 1/12. Nothing about the rod changed; only the axis moved. This is the parallel axis theorem in action: shifting the axis by half the length adds M(L/2)² = ML²/4, and 1/12 + 1/4 is 1/3 exactly. Whenever a shape appears twice in a table with different fractions, that is what you are looking at.
Enter the numbers above and the result panel also gives the torque needed for one radian per second squared, which is numerically the same as I, plus the rotational kinetic energy the object would hold at 100 rpm.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
SuperGlobalCalculator is independently built and maintained. See how we build and verify our calculators.