Moment of Inertia Calculator
Pick a shape — solid sphere, disk, rod, or hoop — enter its mass and radius (or length), and get the moment of inertia I instantly. A live 3D scene spins the chosen shape and charts show how I scales with mass and radius.
Reviewed by the ToolNestr Editorial Team — July 2026
Two ideas that trip students up
1. Mass distribution matters, not just mass
Same mass and radius, two shapes: the solid disk (I = ½mr²) keeps mass close to the axis, while the hoop (I = mr²) puts it all at the rim — twice the moment of inertia.
2. I grows with the square of radius
A curve of I vs radius at fixed mass — doubling the radius quadruples I, because the r² term dominates the linear mass term.
Moment of inertia graphs
How it works
The core idea in one line: moment of inertia is rotational mass — how much an object resists changes in spin, based on both how much mass it has and how far that mass sits from the axis.
I = (2/5) m r²
solid sphere, about a central axis
I = (1/2) m r²
solid disk/cylinder, about its central axis
I = (1/12) m L²
thin rod, about its centre
I = (1/3) m L²
thin rod, about one end
I = m r²
thin hoop/hollow cylinder, or a point mass at radius r
Each shape has its own formula because the mass distribution relative to the axis differs: a hoop keeps all its mass at radius r (I = mr²), while a solid sphere spreads mass from the centre out (I = ⅖mr², the smallest coefficient of the common shapes). Rods depend on where the pivot sits — the centre (⅟₁₂mL²) or the end (⅓mL²).
Worked example 1 — solid cylinder flywheel
Given: A solid steel disk of mass 8.0 kg and radius 0.25 m spins about its central axis. Find I.
Worked example 2 — rod about its end
Given: A thin uniform rod of mass 1.2 kg and length 0.80 m is swung about a pivot at one end. Find I, and compare to spinning it about its centre.
The end-pivot value is exactly 4× the centre-pivot value (⅓ vs ⅟₁₂ = 4× when the mass distribution is the same rod) — spinning about the end is much harder to start or stop.
Moment of inertia across common shapes
All rows use the same mass m = 1 kg and radius (or length) r = 1 m for direct comparison.
| Shape | Formula | I at m=1kg, r=1mkg·m² |
|---|---|---|
| Solid sphere | I = ⅖mr² | 0.400 |
| Solid disk/cylinder | I = ½mr² | 0.500 |
| Thin rod (about centre) | I = ⅟₁₂mL² | 0.083 |
| Thin rod (about end) | I = ⅓mL² | 0.333 |
| Thin hoop / point mass | I = mr² | 1.000 |
For the same mass and radius, a hoop has the largest I (all mass at radius r) and a rod about its centre has the smallest (mass concentrated near the axis).
Where moment of inertia actually matters
🎡 Flywheels
Flywheels store rotational kinetic energy (KE = ½Iω²) to smooth out power delivery in engines and machinery. Engineers concentrate mass near the rim (closer to a hoop than a solid disk) to maximise I for a given mass, storing more energy at the same spin rate.
⛸️ Figure skaters
A skater pulling their arms in reduces their moment of inertia about the spin axis. Since angular momentum L = Iω is conserved with no external torque, a smaller I means a larger ω — the skater spins dramatically faster without adding any energy.
🏏 Sports equipment
Bats, rackets and golf clubs are designed with specific mass distributions to tune their moment of inertia (often called the "swing weight"). A higher I about the grip resists twisting on off-centre hits but is harder to swing quickly, which is why designs balance the two.
Common misconceptions
"Moment of inertia is just the mass of the object."
I depends on both mass and how that mass is distributed relative to the axis — not mass alone. Two objects with identical mass can have very different I values, and the same object has different I values about different axes.
"A heavier object always has a larger moment of inertia."
Not necessarily. A lighter hoop can have a larger I than a heavier solid sphere of the same radius, because I ∝ r² and the hoop concentrates all its mass at the maximum radius while the sphere spreads mass toward the centre.
"You can use one formula for a shape regardless of the axis."
The formula changes with the axis of rotation. A rod about its centre (⅟₁₂mL²) and the same rod about its end (⅓mL²) differ by a factor of 4 — always match the formula to the actual pivot or axis.
"Moment of inertia has the same units as mass."
I is measured in kg·m² (mass times distance squared), not kg. It behaves like mass in rotational equations (τ = Iα mirrors F = ma) but is a distinct physical quantity with different units.
Formula sources & further reading
The formulas here are standard, traceable to:
- • OpenStax, University Physics Volume 1 — §10.5 "Calculating Moments of Inertia" (free, peer-reviewed). openstax.org
- • Halliday, Resnick & Walker, Fundamentals of Physics — Chapter 10, Rotation, Table of rotational inertias.
- • Serway & Jewett, Physics for Scientists and Engineers — Chapter 10, Rotation of a Rigid Object About a Fixed Axis.
Formulas assume idealised uniform-density shapes (thin rods and hoops, solid uniform spheres and cylinders). Results are rounded for display.
How to use this calculator
Choose the shape
Sphere, disk/cylinder, rod (centre or end), hoop, or a point mass all use different formulas.
Enter mass and size
Rods take a length L; other shapes take a radius r.
Watch it spin
Use the sliders to see how mass and radius change I and the spin in 3D.
Related tools
Frequently asked questions
What is moment of inertia?
Moment of inertia (I) is the rotational equivalent of mass — it measures how hard it is to change an object's angular velocity about a given axis. I = Σmᵢrᵢ² for point masses, and standard formulas exist for common rigid-body shapes. Units are kg·m².
Why does the same object have different I values?
Moment of inertia depends on the axis of rotation, not just the object. A rod spun about its centre (I = ⅟₁₂mL²) has a much smaller I than the same rod spun about one end (I = ⅓mL²), because more of its mass sits farther from the end axis.
Why does radius matter more than mass?
I is proportional to mass but to the square of the distance from the axis (r²). Doubling the mass doubles I, but doubling the radius quadruples it — this is why mass distributed far from the axis (like a hoop) resists spin-up far more than the same mass concentrated near the axis (like a solid sphere).
Which shape has the largest I for a given mass and radius?
A thin hoop or hollow cylinder, with I = mr² — all the mass sits at radius r. A solid sphere has the smallest, I = ⅖mr² = 0.4mr², because its mass is distributed from the centre outward, with much of it closer to the axis.
How is moment of inertia used in real calculations?
It appears in rotational kinetic energy (KE = ½Iω²), angular momentum (L = Iω), and Newton's second law for rotation (τ = Iα) — the same roles that mass, ½mv², p = mv, and F = ma play in linear motion.