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 — more mass is concentrated near the center. This is why bowling balls can roll faster than hollow balls of the same weight: they have lower rotational inertia.
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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.
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