Moment of Inertia Calculator

Moment of inertia measures how strongly an object resists a change in its rotational motion about a chosen axis. It plays the rotational role that mass plays in linear motion. A larger moment of inertia means more torque is needed to produce the same angular acceleration. The value depends not only on total mass but also on how far that mass is distributed from the axis, which is why two objects with equal mass can rotate very differently. This calculator covers five idealized shapes used in physics and engineering. A point mass at radius r and a thin hoop both have moment of inertia equal to mass times radius squared because all mass is treated as being the same distance from the axis. A solid cylinder or disk about its central symmetry axis has one half of that value because much of its mass lies inside the outer radius. A uniform solid sphere about a diameter uses two fifths times mass times radius squared. A uniform thin rod rotating about an axis through its center and perpendicular to its length uses one twelfth times mass times length squared. Use kilograms for mass and meters for radius or length to obtain the SI result in kilogram meter squared. Dimensions must describe the same idealized geometry as the selected formula. The radius for a disk, cylinder, hoop, sphere, or point mass is the perpendicular distance from the rotation axis. The rod formula requires its full end-to-end length. If your source measurements use grams, centimeters, or millimeters, convert them before entering values because squaring a length conversion has a large effect on the answer. Real components can often be modeled as combinations of these basic shapes. Add the moments of inertia of components that share the same axis. If an axis is parallel to but displaced from a center-of-mass axis, apply the parallel-axis theorem by adding mass times the square of the displacement. Hollow cylinders with finite wall thickness and rectangular plates need their own formulas rather than an approximate solid-disk result. Always identify both the geometry and the axis before selecting a formula. Moment of inertia is used with torque equals inertia times angular acceleration, and with rotational kinetic energy equal to one half inertia times angular speed squared. Engineers apply it to flywheels, shafts, gears, rollers, motors, and structural dynamics, while students use it in mechanics problems involving rolling and angular momentum. This tool provides a quick value for standard shapes and displays the exact formula used so the result is easy to audit.