Mass Moment of Inertia Calculator

Mass moment of inertia measures how strongly a body resists a change in rotational motion about a specified axis. It plays the rotational role that mass plays in straight-line motion. A larger value means that more torque is needed to produce the same angular acceleration. The SI unit is kilogram square metre, written kg·m². Both the amount of mass and its distance from the axis matter, so two objects with the same mass can have very different rotational behavior. This calculator covers three common idealized bodies rotating about their central symmetry axis. For a uniform solid cylinder or disk, the moment is one half of mass times radius squared. For a thin hoop, essentially all mass lies at the outer radius, so the coefficient becomes one. For a uniform solid sphere rotating about a diameter, the coefficient is two fifths. These formulas assume uniform density, a rigid body, and the axis described above. Real parts with hubs, holes, spokes, or varying density may require decomposition into simpler shapes or numerical integration. Radius has a squared effect. Doubling radius while holding mass constant multiplies inertia by four, while doubling mass only doubles inertia. This explains why a flywheel stores rotational energy efficiently when material is concentrated near its rim. It also explains why a skater spins faster after drawing their arms inward: changing the distribution of mass changes the moment of inertia, and angular momentum is conserved when external torque is negligible. Use consistent SI values for a direct kg·m² answer. Convert grams to kilograms and millimetres or centimetres to metres before entering dimensions. The result can be combined with torque using torque = I × angular acceleration, or with angular speed using rotational kinetic energy = one half × I × angular speed squared. If the rotation axis does not pass through the center of mass, first calculate the central-axis value and then apply the parallel-axis theorem, I = I center + m × d², where d is the offset between parallel axes. The result is a mathematical model rather than a substitute for a detailed engineering mass-properties analysis. Check the selected geometry and axis carefully, retain enough precision during intermediate work, and account for every component when estimating an assembly. Used with those assumptions in mind, this mass moment of inertia calculator provides a fast, transparent starting point for mechanics, machine design, robotics, flywheel analysis, and physics coursework.