🔵 Moment of Inertia Calculator
Calculate rotational inertia for common rigid body shapes.
Moment of Inertia I
How to Use This Calculator
Select the shape from the dropdown, enter the mass and the relevant dimension (radius, length, or both), and the calculator gives you the moment of inertia in kg·m². The formula for the selected shape is shown dynamically so you can check which measurement goes where.
Pick the shape that best matches your object. A wheel is typically a solid cylinder (I = ½MR²), while a bicycle tyre approximates a hollow thin-wall cylinder (I = MR²).
Enter the total mass M in kilograms. For a 2 kg steel disk with 0.15 m radius, type 2 in the mass field.
Enter the required dimension. For cylinders and spheres this is the radius R. For rods it is the length L.
Read the result in kg·m². Use it in rotational equations like τ = Iα or KE = ½Iω².
Moment of Inertia Formulas
M is mass in kg, R is radius in metres, and L is length in metres. The parallel axis theorem lets you shift any formula to a different rotation axis: just add M × d², where d is the distance between the new axis and the centre-of-mass axis.
Worked Examples
Where This Comes Up in Real Life
When a figure skater pulls their arms in, they reduce their moment of inertia. Because angular momentum is conserved, their spin speed increases. A skater spinning with arms out at I = 4 kg·m² at 1 rev/s who tucks to I = 1 kg·m² will spin at 4 rev/s. You can calculate this exactly by using I₁ω₁ = I₂ω₂.
Flywheels in engines store rotational kinetic energy to smooth out power pulses. A solid steel flywheel of mass 20 kg and radius 0.15 m has I = ½ × 20 × 0.0225 = 0.225 kg·m². Spinning at 3000 RPM (314 rad/s), it stores KE = ½ × 0.225 × 314² = 11,120 J, about the energy needed to accelerate a car from 0 to 15 km/h. Hollow flywheels store more energy per kilogram because more mass sits at the outer edge.
Frequently Asked Questions
What is moment of inertia?
Moment of inertia I is the rotational analog of mass. It measures an object's resistance to changes in rotation. I = Σmr², where r is the distance of each mass element from the axis.
Why does shape matter for moment of inertia?
Objects with mass concentrated farther from the rotation axis have a higher I. A hollow cylinder has higher I than a solid cylinder of the same mass and radius.
What is the parallel axis theorem?
I = I_cm + M × d², where I_cm is the moment of inertia about the center of mass, M is total mass, and d is the distance between axes. Allows calculation about any parallel axis.
What are the units of moment of inertia?
kg·m² in SI. Moment of inertia depends on both mass and the square of the distance from the axis.
Why do solid spheres roll faster than hollow ones?
Hollow sphere has I = ⅔MR² vs solid sphere I = ⅖MR². Higher I means more energy goes into rotation, leaving less for translation — so it rolls slower.