Moment of Inertia Calculator
Find the moment of inertia for common shapes.
Results update as you type.
How it works
The moment of inertia is the rotational equivalent of mass: it measures how strongly an object resists being spun up or slowed down about an axis, and this calculator computes it for the most common shapes. You choose a shape, such as a solid sphere, a cylinder, a hoop, or a rod, and enter the mass and the relevant radius or length; the calculator applies the standard formula for that shape, each of which is a coefficient times mass times the square of the distance. The key insight is that where the mass sits matters as much as how much there is: mass located far from the axis contributes far more to the moment of inertia than mass near it, because of the squared distance. That is why a hollow hoop resists spinning twice as much as a solid disk of equal mass and radius, and why figure skaters spin faster when they pull their arms in, reducing their moment of inertia. This quantity governs angular acceleration through the rotational form of Newton's second law, and it appears in the energy and momentum of anything that rotates, from flywheels and gears to planets and molecules. Engineering and physics students use it constantly. Pick a shape and enter the two values to get the moment of inertia.
I = k · m · d² (k depends on the shape)
The moment of inertia measures an object's resistance to rotational acceleration and equals a shape-dependent coefficient times the mass times the square of a characteristic distance. Concentrating mass far from the axis raises the moment of inertia, which is why a hoop, with all its mass at the rim, has twice the value of a solid disk of the same mass and radius. The coefficient captures how the mass is distributed relative to the rotation axis.
Source:Rotational dynamics (standard shape formulas)
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter shape, mass (m) and radius or length.
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Moment of inertia updates instantly as you type — no button to press.
Frequently asked questions
What is moment of inertia?
Moment of inertia is a measure of an object's resistance to changes in its rotation about a given axis, playing the same role for rotation that mass plays for straight-line motion. It depends on both the total mass and how that mass is distributed relative to the axis. The farther the mass lies from the axis, the greater the moment of inertia and the harder the object is to spin up.
Why does the shape matter?
Because moment of inertia depends on how mass is distributed, not just how much there is, different shapes with the same mass and size have different values. A solid disk has half the moment of inertia of a hoop of equal mass and radius, because the disk's mass is spread inward while the hoop's is all at the rim. Each shape has its own coefficient in the formula.
What is the difference between radius and length inputs?
For round shapes like spheres, cylinders, hoops, and point masses, the relevant distance is the radius from the axis. For a thin rod, it is the full length of the rod, and the formula differs depending on whether it rotates about its centre or about one end. Choose the shape that matches your object, then enter the appropriate radius or length.
Why do skaters spin faster with arms tucked in?
Pulling the arms inward moves mass closer to the rotation axis, which lowers the moment of inertia. Because angular momentum is conserved when no external torque acts, a smaller moment of inertia means a faster spin. The same principle lets divers and gymnasts control their rotation speed by changing their body shape in mid-air.
How is moment of inertia used?
It determines how much torque is needed to achieve a given angular acceleration, through the rotational version of Newton's second law. Engineers use it to design flywheels, gears, motors, and rotating machinery, and it appears in the kinetic energy and angular momentum of anything that spins. Astronomers use it to study the rotation of planets and stars, and it is central to any analysis of rotational dynamics.