Moment of Inertia Calculator
Compute the rotational inertia of a point mass, thin ring, solid disk, solid sphere or rod about its centre.
How it works
Point / ring: I = m r² · Disk: I = ½ m r² · Sphere: I = ⅖ m r² · Rod (centre): I = m L² ÷ 12
Moment of inertia is rotation’s answer to mass: it measures how hard it is to change a body’s spin, and it depends on how the mass is distributed relative to the axis. Mass placed far from the axis counts far more because the radius is squared, which is why a hollow ring resists spinning more than a solid disk of the same mass and radius. Each standard shape has a fixed coefficient in front of m r².
Worked example
A solid disk of 2 kg and radius 0.3 m has I = ½ × 2 × 0.3² = ½ × 2 × 0.09 = 0.09 kg·m². A thin ring of the same mass and radius would be twice that, 0.18 kg·m².
Frequently asked questions
Why does a ring have more inertia than a disk?
All the ring’s mass sits at the full radius, whereas a disk spreads mass from the centre outward. Since inertia scales with r², mass concentrated at the rim contributes the maximum, giving the ring twice the coefficient of a disk.
What are the units of moment of inertia?
Kilogram-metre-squared (kg·m²) in SI. It combines a mass in kg with a squared distance in metres.
What axis do these formulas assume?
The disk, ring, sphere and point formulas are about the central symmetry axis; the rod formula is about an axis through its centre, perpendicular to its length.
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This calculator is for educational purposes. Double-check important results before acting on them.