Moment of Inertia Calculator (common shapes)

Understanding the moment of inertia calculator — concept and principle

The moment of inertia I measures how hard it is to change an object's rotation, the rotational counterpart of mass. It depends not only on how much mass there is but on how far that mass sits from the axis. It matters for flywheels, wheels, shafts and rolling objects. This calculator applies the standard formulas for six uniform shapes.

Where the mass sits matters: the default solid disc gives 0.25 kg·m², half the ring’s.

How it is calculated

I = k × m × r², where m is the mass in kg, r is the radius (or, for rods, the length L) in m, and k is a shape factor:

Solid disc or cylinder about its central axis: k = 1/2
Thin ring or thin-walled cylinder: k = 1
Solid sphere about a diameter: k = 2/5
Thin hollow sphere: k = 2/3
Thin rod about its centre, perpendicular to the rod: k = 1/12 (with L instead of r)
Thin rod about one end: k = 1/3

The result is in kg·m².

The principle

Each small piece of mass dm contributes r² dm, so I = ∫ r² dm. Mass far from the axis counts much more because of the square. A ring has all its mass at the rim, so k = 1, the largest possible for a given outer radius. A solid disc has much of its mass near the centre and gets k = 1/2. A sphere concentrates even more mass close to the axis and has k = 2/5. For a rod, moving the axis from the centre to one end raises I from 1/12 to 1/3 mL²; the parallel-axis theorem I = I_cm + m d² explains it: 1/12 + (1/2)² = 1/3.

Worked example

Inputs: m = 2 kg, r = 0.5 m, shape = solid disc (k = 1/2).
r² = 0.5 × 0.5 = 0.25 m².
I = 0.5 × 2 × 0.25 = 0.25 kg·m².

Each point is the calculator re-run with only “Radius / length (m)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Radius / length (m)	I (kg·m²)
0.125	0.015625
0.25	0.0625
0.375	0.140625
0.5	0.25
0.625	0.390625
0.75	0.5625
1	1

How to read the result

A larger I means more torque is needed for the same angular acceleration (τ = Iα) and more energy is stored at the same speed (E = ½Iω²). With the same 2 kg and 0.5 m, a thin ring would give 0.5 kg·m², twice the disc, while a solid sphere would give 0.2 kg·m². This is why a solid ball rolls down a slope faster than a hoop of the same mass and radius. Flywheels put mass at the rim to store energy; fast-turning parts keep mass near the axis to accelerate quickly.

Common mistakes

Using a diameter instead of a radius, which makes I four times too big.
Choosing a rod formula but entering half the length as if it were a radius.
Applying a formula for a different axis; I always belongs to a specific axis.

Keep in mind: The formulas assume uniform density and ideal shapes (perfectly thin rings, rods and shells). Real parts with hubs, spokes or holes need the parts added up, CAD software or measurement, and critical rotating machinery should be checked by a qualified engineer.

Frequently asked questions

What is the radius of gyration?

It is the distance k_g at which all the mass could be placed to give the same I, so I = m k_g². For a solid disc k_g = r ÷ √2 ≈ 0.707r.

How do I find I about a different axis?

Use the parallel-axis theorem: I = I_cm + m d², where d is the distance between the new axis and a parallel axis through the centre of mass.

How do I handle a hollow thick cylinder?

For inner radius r₁ and outer radius r₂, I = ½ m (r₁² + r₂²) about the central axis.

Sources: Halliday, Resnick and Walker, Fundamentals of Physics, table of rotational inertia · J. L. Meriam and L. G. Kraige, Engineering Mechanics: Dynamics, appendix on mass moments of inertia

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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