Moment of Inertia Calculator
Rotational inertia for six shapes, metric and imperial.
What it does
Moment of inertia is mass with a memory of distance: the farther each bit sits from the axle, the harder the spin. Hoops out resist disks, and rod ends out resist centers, at equal mass.
The tool above takes shape, mass, and radius or length in metric or imperial. It returns rotational inertia with the shape factor.
The default 2 kg disk at 0.5 m gives 0.25 kg m2. The imperial mirror at 4.41 lb with 1.64 ft reads 5.93 slug ft2.
University physics tables fix the six factors used here, from hoop 1 down to rod center 1/12. LibreTexts rotation chapters repeat the same table for fixed axis drills.
Inertia feeds everything spinning: torque needs it for angular acceleration, and flywheels bank energy as half I omega squared. Size the I here, then spend it in those pages.
How to use
- Pick metric or imperial.
- Pick the shape.
- Type mass.
- Type radius or length.
- Read inertia first.
- Read the factor beside it.
Formula
- m
- Body mass in kg or lb.
- R, L
- Radius or rod length in m or ft.
- I
- Shape factor times mR2 or mL2.
OpenStax University Physics tabulates the same six factors in its rotation chapter, and LibreTexts physics mirrors the table for fixed axis problems. Imperial converts mass and length to SI first, then reports slug ft2 at 23.7304 per kg m2. Flywheel energy pages consume this I inside 1/2 I w2.
Worked example
Example: 2 kg disk at 0.5 m
Metric defaults: disk, 2 kg, 0.5 m.
- Body: solid disk, 2 kg, 0.5 m radius.
- Factor: 1/2 x mR2 for disks.
- Math: 0.5 x 2 x 0.25 = 0.25 kg m2.
- Imperial: 4.41 lb with 1.64 ft reads 5.93 slug ft2.
- Spin: inertia resists angular change.
- Tool shows 0.25 kg m2, 1/2 x mR2, Resists angular change.
Reference table
| Shape | Factor | At 1 kg, 1 m |
|---|---|---|
| Hoop / point | 1 | 1 kg m2 |
| Solid disk | 1/2 | 0.5 kg m2 |
| Solid sphere | 2/5 | 0.4 kg m2 |
| Rod, end | 1/3 | 0.33 kg m2 |
| Rod, center | 1/12 | 0.083 kg m2 |
Limits
- Symmetric solids about listed axes.
- Rigid bodies; flex adds modes.
- Fixed axis; tumbling differs.
- Point mass needs true small size.
Common mistakes
- Using diameter as radius.
- Forgetting the 1/2 on disks.
- Mixing rod center with end.
- Unit hops between kg and lb.
- Squaring after converting wrong.
Try it yourself
Quick checks. Answers sit tucked under each question.
Hoop, 1 kg at 1 m?
1 x 1 x 1 = 1 kg m2.
Rod center, 12 kg by 1 m?
12 x 1 / 12 = 1 kg m2.
Mass 0?
Zero mass spins nothing: enter mass above 0.
Questions
Shape factors?
Hoop 1, disk 1/2, rod center 1/12, rod end 1/3, sphere 2/5, point 1, each times mR2 or mL2.
Why placement matters?
Mass far from the axis counts most: hoops beat disks, rod ends beat centers.
Torque link?
Torque equals inertia times angular acceleration. Flywheels store 1/2 I w2 on the same I.
Units?
kg m2 metric, slug ft2 imperial. Multiply kg m2 by 23.7304.
Odd shapes?
No, this page covers symmetric solids. Composite bodies add piece by piece.
Spin up faster?
Yes, through the same I: double speed needs double torque at fixed I.