Mass Moment of Inertia Calculator
Select a rigid-body shape, enter the mass and dimensions, and get the rotational inertia (I) about the standard axis - instantly and with full working shown. Supports metric (kg, m) and imperial (lb, ft/in) units, includes the parallel-axis theorem for off-center axes, and covers ten common geometries used in physics and engineering.
What is the mass moment of inertia?
The mass moment of inertia (symbol I, also called rotational inertia or angular mass) measures how strongly a rigid body resists angular acceleration about a given axis. It plays the same role in rotational dynamics that ordinary mass plays in linear dynamics: Newton's second law for rotation is torque = I * alpha, where alpha is angular acceleration. The larger I is, the more torque is required to spin the object up to a given rotational speed. Because I depends on how mass is distributed relative to the axis, two objects with the same total mass can have very different rotational inertias depending on their shape and orientation.
How to use this calculator
Choose a unit system (metric or imperial), then select the shape of your object from the dropdown. Enter the mass and the relevant dimension - radius for spheres, disks and cylinders; length for rods; width and height for rectangular plates. For a hollow cylindrical tube, an inner radius field appears automatically. If you need the moment about an axis that does not pass through the centroid, enable the parallel-axis theorem toggle and enter the perpendicular distance d from the centroidal axis to your target axis; the calculator adds m*d^2 automatically. The result panel shows the total I, the centroidal I before the shift, the parallel-axis correction term, and a description of the axis. The "Show your work" panel traces every arithmetic step with your actual numbers.
The parallel-axis theorem
The parallel-axis theorem states that I = I_cm + m * d^2, where I_cm is the moment of inertia about the centroidal axis (passing through the center of mass), m is the total mass, and d is the perpendicular distance between the centroidal axis and the new parallel axis. Because inertia grows with d^2, shifting the axis even a small distance from the center of mass can noticeably increase I. For example, a 2 kg solid sphere with radius 0.2 m has I_cm = 0.032 kg*m^2. Shifting the axis 0.5 m adds m*d^2 = 2 * 0.25 = 0.5 kg*m^2, more than fifteen times the centroidal value. This theorem is indispensable in engineering: it lets you combine the moments of sub-components about their own centroids and then shift each to a common reference axis to get the total system inertia.
Shape factors and practical examples
Each shape has a characteristic shape factor that determines how efficiently its mass is placed relative to the axis. A circular hoop has a shape factor of 1 (all mass at radius r), a solid disk has 0.5, a solid sphere has 0.4, and a thin rod rotating about its center has 1/12 of its length squared. In practice, a flywheel designer wants a high shape factor (concentrated at the rim) to store maximum energy at a given angular speed. A spinning top or gyroscope benefits from a large I to maintain stability. An engine crankshaft must have a carefully calculated I to limit torsional vibration. Knowing I also lets you compute rotational kinetic energy as KE = (1/2) * I * omega^2, where omega is angular velocity in radians per second.
Moment of inertia formulas for common rigid bodies
| Shape | Axis | Formula for I_cm |
|---|---|---|
| Solid sphere | Any through center | I = (2/5) m r^2 |
| Thin spherical shell | Any through center | I = (2/3) m r^2 |
| Solid cylinder | Symmetry (central) axis | I = (1/2) m r^2 |
| Hollow cylinder (tube) | Symmetry axis | I = (1/2) m (r_o^2 + r_i^2) |
| Thin disk | Axis perp. to face, center | I = (1/2) m r^2 |
| Circular hoop | Axis perp. to plane, center | I = m r^2 |
| Thin rod | Axis through center, perp. | I = (1/12) m L^2 |
| Thin rod | Axis through one end, perp. | I = (1/3) m L^2 |
| Rectangular plate | Axis perp. to face, center | I = (1/12) m (a^2 + b^2) |
| Solid cone | Symmetry axis (apex to center) | I = (3/10) m r^2 |
I_cm = moment of inertia about the centroidal axis. Use the parallel-axis theorem (I = I_cm + m*d^2) to shift to any parallel axis.
Frequently asked questions
What is the difference between mass moment of inertia and area moment of inertia?
The mass moment of inertia (units: kg*m^2 or lb*ft^2) measures resistance to rotational acceleration and involves the actual mass of the object. The area moment of inertia (second moment of area, units: m^4 or in^4) is a purely geometric property of a cross-section, used in beam-bending calculations to relate bending moment to stress and deflection. They are different quantities with different units, although both are called "moment of inertia" in some contexts. This calculator computes the mass moment of inertia.
How does changing the axis affect the moment of inertia?
Moving the rotation axis away from the center of mass always increases the moment of inertia, by exactly m*d^2 (the parallel-axis theorem). The centroidal axis gives the minimum possible I for any parallel orientation. This means a rod rotating about its end has four times the inertia of the same rod rotating about its center, because the average squared distance of the mass from the axis is four times larger.
What units should I use?
The SI unit is kilogram meter squared (kg*m^2). In the imperial system, the commonly used units are pound-foot squared (lb*ft^2) or pound-foot-second squared (lb*ft*s^2, used when weight rather than mass is the input). This calculator uses lb and ft for inputs and lb*ft^2 for the output in imperial mode, with an exact conversion factor of 1 kg*m^2 = 23.7304 lb*ft^2.
Can I compute the moment of inertia for a composite object?
Yes, but you must do it in parts. Compute I for each sub-component about the shared reference axis (using the parallel-axis theorem to shift from each centroid to the common axis if needed), then sum the results. For example, a barbell is two spheres plus a rod: compute I_rod about its center, compute I_sphere for each ball using the parallel-axis theorem to shift to the barbell's symmetry axis, then add all three. This calculator handles one shape at a time; add the results from multiple runs.
What is the relationship between moment of inertia and torque?
Torque (tau), moment of inertia (I), and angular acceleration (alpha) are related by tau = I * alpha, the rotational analogue of F = m * a. If you apply a constant net torque of 10 N*m to an object with I = 2 kg*m^2, it accelerates at alpha = 10/2 = 5 rad/s^2. A larger I means a smaller angular acceleration for the same torque, just as a heavier object accelerates more slowly under the same force.