Polar Moment of Inertia Calculator

Use this tool to calculate the polar moment of inertia of a solid or hollow circular section.

Moment of inertia0.015 kg·m²0.5 × 12 kg × (0.05 m)²

ShapeSolid disc or cylinderabout its central axis

Coefficient k0.5the fraction of mr² this shape and axis gives

Radius of gyration0.035355 mthe radius at which a point mass would have the same moment of inertia

As a point mass at a distance0.03 kg·m²2× yours, at the same mass and size

As a thin hoop or ring0.03 kg·m²2× yours, at the same mass and size

As a thin hoop, about a diameter0.015 kg·m²1× yours, at the same mass and size

As a solid cylinder, about a diameter0.0075 kg·m²0.5× yours, at the same mass and size

As a solid sphere0.012 kg·m²0.8× yours, at the same mass and size

As a thin spherical shell0.02 kg·m²1.3333× yours, at the same mass and size

As a solid cone0.009 kg·m²0.6× yours, at the same mass and size

The formula

I = kmr², with k depending on the shape and axis

Where the mass sits is what matters

Moment of inertia is rotational mass: it measures how hard a body is to spin up, and it depends not just on how much mass there is but on how far that mass sits from the axis. The r² is why — mass at twice the radius contributes four times as much.

That is why a hoop, with all its mass at the rim, has exactly twice the moment of inertia of a solid disc of the same mass and radius, and why a solid sphere is lower still at two fifths. Race down a slope and they finish in that order reversed: the sphere wins because less of its energy goes into spinning.

One body has a different moment about every axis, so the axis is part of the answer, not a footnote. A rod about its end has four times the moment it has about its centre. The parallel-axis theorem relates them: shifting the axis a distance d from the centre of mass adds md², which means the centre of mass is always the axis of least resistance.