Moment of Inertia CalculatorRotational inertia · I = kmr²
Moment of inertia (I) measures how hard it is to change an object’s rotation. For any shape it is I = k·m·r², where k depends on how the mass is spread out. Pick a disc, ring, sphere, rod or plate and the axis, then enter the mass and size. You get I in kg·m², the radius of gyration and the rotational energy at 1,000 rpm.
Formula verified against OpenStax — University Physics, Volume 1 · HyperPhysics, Georgia State University
By Manzoor Ahmad · Pharmacist & calculator developerReviewed & updated
Shape & dimensions
Choose a shape family, then the exact form and axis.
Step-by-step working
What is moment of inertia?
Moment of inertia is the rotational equivalent of mass: it tells you how much torque you need to give an object a certain angular acceleration. It depends on the total mass and on how far that mass sits from the axis of rotation.
Flywheels, figure skaters, gymnasts, rolling balls and spinning hard drives all come down to moment of inertia. Engineers put mass at the rim when they want rotation to be stable and hard to change.
Quick answers
How do you calculate the moment of inertia of a disc?
For a solid disc about its centre, I = ½mr². A 5 kg, 0.2 m-radius disc has I = 0.5 × 5 × 0.04 = 0.1 kg·m². Choose Disc in the first tab.
What is the moment of inertia of a rod about its end?
I = ⅓mL², four times larger than about its centre (¹⁄₁₂mL²). A 1 m, 1 kg rod gives 0.333 kg·m² about its end.
What is the radius of gyration?
The distance k at which all the mass could be concentrated to give the same inertia: k = √(I ÷ m). For a solid disc k ≈ 0.707r; for a solid sphere k ≈ 0.632r.
Why does a solid ball roll faster down a slope than a hollow one?
A solid sphere has less inertia (⅖mr² against ⅔mr²), so less of its energy goes into spinning and more into forward speed. Compare with an inclined plane block that slides without rolling.
Moment of inertia formulas
Shift the axis with the parallel-axis theorem: I = Icm + md².
| Symbol | Meaning | SI unit |
|---|---|---|
| I | Moment of inertia | kg·m² |
| m | Mass | kg |
| r, L | Radius or length | m |
| k | Shape factor (½ for a disc, ⅖ for a solid sphere…) | — |
| ω | Angular velocity | rad/s |
| α | Angular acceleration | rad/s² |
How to calculate moment of inertia
- Identify the shape and the axis; the same object has different I about different axes.
- Look up the shape factor k (see the table below).
- Multiply k × mass × radius² (in metres).
- For an off-centre axis, add md² (parallel-axis theorem).
Worked example: a 2 kg solid disc
A 2 kg solid disc of radius 10 cm spins about its central axis.
At 1,000 rpm (104.7 rad/s): KE = ½ × 0.010 × 104.7² = 54.8 J
A thin ring of the same mass and radius has twice the inertia (0.020 kg·m²), because all its mass sits at the rim.
Moment of inertia of common shapes
About the axis described, through the centre of mass unless noted.
| Shape | Axis | I |
|---|---|---|
| Point mass | Distance r | mr² |
| Thin ring / hoop | Central, perpendicular | mr² |
| Solid disc / cylinder | Central, along length | ½mr² |
| Solid disc | Through a diameter | ¼mr² |
| Solid sphere | Through centre | ⅖mr² |
| Thin spherical shell | Through centre | ⅔mr² |
| Thin rod | Through centre | ¹⁄₁₂mL² |
| Thin rod | Through one end | ⅓mL² |
| Rectangular plate | Perpendicular through centre | ¹⁄₁₂m(a² + b²) |
Common mistakes to avoid
- Forgetting to square the radius. I scales with r², so doubling the size quadruples it.
- Using the diameter as r. Halve it first.
- Ignoring the axis. A rod about its end has four times the inertia it has about its centre.
Manzoor Ahmad
Pharmacist turned Web Developer & Calculator Engineer · 13+ years in web developmentI’m a pharmacist by training who turned into a full-time tech geek, and for 13+ years I’ve designed and coded web tools. My pharmacy degree included physical pharmacy and pharmaceutics — states of matter, heat, flow and reaction kinetics — so the physics behind these tools is familiar ground, not a new hobby.
I’m not a physicist, so I don’t ask you to take my word for it: every formula on this page is traced to an open university textbook or NIST reference data, and the calculator is tested against published worked examples before it goes live. Spot an error? Tell me and it will be fixed and dated on the next update.
Moment of Inertia Calculator: frequently asked questions
It is the rotational equivalent of mass: a measure of how strongly an object resists changes to its rotation about a particular axis. It is measured in kg·m².
For point masses, I = Σmr². For standard shapes, I = k·m·r², with k = ½ for a solid disc, ⅖ for a solid sphere and ¹⁄₁₂ for a rod about its centre.
I = Icm + md², where d is the distance between the new axis and a parallel axis through the centre of mass.
No. Mass moment of inertia (kg·m²) is about rotation dynamics. The second moment of area (m⁴), also loosely called ‘moment of inertia’, is about beam bending.
Pulling mass inward reduces I. With angular momentum L = Iω conserved, ω has to rise.
References & sources
- OpenStax — University Physics, Volume 1 — Ch. 10.4–10.5, Moment of Inertia
- HyperPhysics, Georgia State University — moment of inertia of common shapes
- OpenStax — College Physics 2e — Ch. 10.3, Dynamics of Rotational Motion
Disclaimer: This calculator is an educational tool built on idealised textbook models. It is not a substitute for professional engineering or safety calculations.