Mass Moment of Inertia (Shapes)
Rotational inertia for the four shapes that cover most machine parts.
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The engineering
Moment of inertia is mass weighted by radius *squared* — where the material sits matters far more than how much there is. That's why the hollow cylinder's formula adds the radii instead of subtracting: for equal mass, a tube always out-inertias a solid disc, and flywheel rims live at the largest radius the stress allows.
The radius of gyration row is the intuition cheat: it's the single radius at which all the mass could sit and match the real I. Motor sizing is the everyday customer — the reflected inertia a servo fights scales with these numbers, and gearboxes divide it by ratio squared, which is why a modest gearhead tames a monstrous load.
Where this math comes from
Christiaan Huygens needed the concept before it had a name: his 1673 'Horologium Oscillatorium' solved the compound pendulum — how a swinging *body*, not a point mass, keeps time — effectively computing mass-distribution integrals to make clocks honest. It was rotational dynamics done bare-handed, twenty years before Principia.
Leonhard Euler named and systematized the quantity in his 1765 'Theoria motus corporum solidorum', defining the moment of inertia, its principal axes, and the equations of rigid-body rotation that everything from turbine rotors to tumbling spacecraft still obeys. The standard tables of shapes on this card are Euler's program filled in.
- 1673Christiaan HuygensCompound pendulum solved — rotational inertia in practice.
- 1765Leonhard Euler'Moment of inertia' defined; rigid-body dynamics founded.
- 1834Louis PoinsotGeometric picture of free rotation — the inertia ellipsoid.
See the full timeline of the math behind every calculator →
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