Point Mass Moment of Inertia Calculator

Enter the mass in kilograms and its perpendicular distance from the axis in metres. The result is returned in kilogram metre squared, written kg·m². It is designed for particles, concentrated-mass models, and systems where an object's dimensions can be neglected relative to its distance from the axis. The calculation is deterministic, runs without network access, and rejects negative mass rather than producing a physically misleading answer. API requests cost $0.002, while the browser calculator uses the same calculation logic.

Understand the point-mass model and its axis

Moment of inertia describes how strongly mass resists a change in rotational motion about a chosen axis. For one ideal point mass, the entire mass is treated as if it occupies a single location, so the relevant distance is the shortest perpendicular distance from that location to the axis. The formula is I = mr², where m is mass and r is that perpendicular distance. Squaring the radius makes position especially important: moving a mass twice as far from the axis produces four times the moment of inertia, even when its mass stays unchanged. The axis must therefore be identified before choosing a radius. A point may be several metres from an arbitrary reference origin but have a much smaller perpendicular distance to the actual rotation axis. This calculator is appropriate when the object's physical size is negligible for the problem or when a larger assembly has already been simplified into concentrated masses. It does not calculate the distributed inertia of a disc, rod, sphere, or other extended body. For those objects, use the formula matching their geometry and specified axis.

Enter SI values and interpret the result

Provide mass as a finite number in kilograms and radius as a finite number in metres. Both values may be zero, but neither may be negative. A zero mass has zero moment of inertia, and a point located directly on the axis also has zero moment of inertia because its perpendicular radius is zero. The solver multiplies mass by radius and radius again, then returns moment_of_inertia in kg·m² together with the inputs and the formula used. For example, a 3 kg point mass located 2 m from an axis has I = 3 × 2² = 12 kg·m². Convert measurements before calling the calculator if your source uses grams, centimetres, pounds, or feet; the endpoint does not infer or convert units. A radius of 50 centimetres must therefore be entered as 0.5 metres. The result is a scalar for the axis implied by the supplied perpendicular distance. It is not a full inertia tensor, and it does not determine angular velocity, torque, angular momentum, or rotational energy without additional quantities and equations.

Apply the calculation to systems and checks

A single point-mass result is often a building block for a larger mechanics calculation. For several discrete masses rotating about the same axis, call the relationship for each mass and add the individual values: total inertia is the sum of mᵢrᵢ². Keep every mass in kilograms and every perpendicular radius in metres so all terms share kg·m² before addition. The same expression appears in the parallel-axis theorem, where md² accounts for shifting an object's known centre-of-mass inertia to a parallel axis, although this calculator supplies only the point-mass term. You can also use its result in L = Iω for angular momentum, τ = Iα for constant-axis rotational dynamics, or E = ½Iω² for rotational kinetic energy when the assumptions behind those scalar equations apply. As a practical check, moment of inertia cannot be negative for valid mass and radius, and increasing either positive input cannot reduce the answer. Confirm that radius was measured perpendicular to the axis, not along a curved path or from an unrelated origin. The API uses the same deterministic solver as the browser tool and charges $0.002 for a successful request.