How Moment of Inertia Is Calculated
Moment of inertia (also called rotational inertia) measures how hard it is to change a body's rotation about a given axis. For a collection of point masses it is I = Σm_i r_i², and for a continuous rigid body it is the integral I = ∫r² dm, where r is each mass element's perpendicular distance from the axis. It is the rotational analog of mass: in the rotational version of Newton's second law, torque = I × angular acceleration, a larger I means more torque is needed to produce the same angular acceleration. Crucially, I depends not just on total mass but on how far that mass sits from the axis — the same mass arranged closer to the axis gives a smaller I, and arranged farther out gives a larger I.
Formula and method
This calculator uses the standard closed-form moment-of-inertia formulas for common rigid bodies, each about an axis through the center of mass: a solid sphere uses I = (2/5)MR²; a thin spherical shell uses I = (2/3)MR²; a solid cylinder or disk spinning about its central longitudinal axis uses I = (1/2)MR²; a thin hoop or ring uses I = MR²; a thick-walled tube (outer radius R, inner radius r) uses I = (1/2)M(R² + r²); a thin rectangular plate about the axis perpendicular to its face uses I = (1/12)M(a² + b²); and a thin rod about a perpendicular axis through its center uses I = (1/12)ML². Enter your mass and dimensions in any supported unit — the calculator converts them internally to kilograms and meters, computes I_cm in kg·m² (the SI unit for moment of inertia), and if you enter a nonzero axis offset distance d, applies the parallel axis theorem I = I_cm + Md² to find the moment of inertia about that shifted axis. It also reports the radius of gyration, k = √(I_cm / M), the distance from the axis at which all the mass could be concentrated as a point and still produce the same I_cm.
Common sources of error
- Radius vs. diameter: all these formulas use radius R, not diameter D — if you measured across the whole object, divide by 2 first (R = D/2).
- Wrong axis convention: the same object has a different moment of inertia depending on the chosen axis — a disk spinning about its central spin axis uses I = ½MR², but the same disk tipped to rotate about a diameter uses I = ¼MR², a very different number.
- Mixing units: keep mass and length consistent (this calculator converts your chosen units internally, but double-check that the mass unit and length unit selects match your actual measurements).
- Measuring d from the wrong point: the parallel axis theorem only works when d is measured from the object's own center-of-mass axis to the new axis — not between two arbitrary axes.
Checking your result
A few quick sanity checks: for the same mass and radius, a hoop (I = MR²) always has a larger moment of inertia than a solid disk (I = ½MR²), which in turn is larger than a solid sphere (I = ⅖MR²) — because the hoop concentrates mass farthest from the axis. Moment of inertia also scales with the square of the size dimension (doubling R quadruples I for a given shape) and scales linearly with mass (doubling M doubles I). If your result does not move in these directions when you change an input, recheck which shape and axis you selected.
Applications
Moment of inertia is central to any rotating-machinery or motion problem: engineers use it to size motors and flywheels (torque = I × angular acceleration, so a larger I needs more torque to spin up or brake in a given time), to store rotational kinetic energy (KE = ½Iω²) in flywheels, and to select actuators for robot arms and joints. Figure skaters and divers pull their arms and legs inward to reduce I and spin faster, since angular momentum L = Iω is conserved when no external torque acts — a smaller I forces a larger ω. Vehicle wheels, satellites, sports equipment (bat and racket "swing weight"), and gyroscopes are all designed around their moment of inertia about specific axes.