Disk Moment of Inertia Calculator
A uniform solid disk or cylinder rotating about its central axis has a moment of inertia of one half the mass times the radius squared. This single quantity captures how hard it is to angularly accelerate the disk and how much energy it stores when spinning. This calculator takes the disk's mass, its radius, and an angular speed, then returns the moment of inertia about the central axis, the radius of gyration, and the rotational kinetic energy. The result is exact for a uniform disk of any thickness.
Disk moment of inertia formula
Moment of inertia I = 0.5 * m * r^2
Radius of gyration k = r / sqrt(2)
Rotational kinetic energy = 0.5 * I * omega^2
The factor of one half comes from integrating the mass over the disk: mass closer to the axis contributes less. The radius of gyration is the equivalent point-mass radius, and the kinetic energy uses angular speed in radians per second.
Rotational inertia facts
- A solid disk has I equal to one half m r squared about its central axis.
- A thin hoop of the same mass and radius has twice the moment of inertia.
- Thickness does not change the central-axis moment of inertia.
- The radius of gyration of a disk is about 0.707 times its radius.
- Rotational kinetic energy rises with the square of angular speed.
Disk moment of inertia: frequently asked questions
What is the moment of inertia of a solid disk?
A uniform solid disk or cylinder rotating about its central axis has a moment of inertia of one half the mass times the radius squared: I = 0.5 m r squared. This is a standard result from integrating the mass distribution and applies regardless of the disk's thickness.
What is the radius of gyration?
The radius of gyration k is the distance from the axis at which a point mass equal to the total mass would have the same moment of inertia. For a solid disk it is the radius divided by the square root of 2, about 0.707 times the radius, because I equals m k squared.
How is rotational kinetic energy calculated?
Rotational kinetic energy is one half the moment of inertia times the angular velocity squared: 0.5 I omega squared, with angular velocity in radians per second. This calculator computes it from the disk's moment of inertia and an angular speed you enter.
Does disk thickness affect the moment of inertia?
No, not for rotation about the central axis. A solid cylinder of any length has the same one half m r squared about its axis, because every slice has the same radial mass distribution. Thickness matters only for rotation about a different axis.
How does this differ from a hoop or ring?
A thin hoop has all its mass at the rim, giving I = m r squared, twice that of a solid disk of the same mass and radius. The solid disk's factor of one half reflects that much of its mass sits closer to the axis where it contributes less to the moment of inertia.
Official sources
- NIST: Physical Measurement Laboratory: mechanics and rotational quantities.
- NASA: NASA Glenn Research Center: rotational dynamics.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.