Moment of Inertia Calculator
Get the rotational inertia of the standard textbook shapes. Enter mass (kg, g, or lb) and radius or rod length (m, cm, in, or ft), pick the shape, and read I in kg·m² and lb·ft² with the exact formula used.
Example: with Shape and axis Solid sphere — I = (2/5)·m·r² · Mass 2 · Mass unit kg · Radius r (or rod length L) 0.5 · Length unit m (meters) → Moment of inertia I: 0.2 kg·m².
- How it's computedI = 2/5 × 2 kg × (0.5 m)² = 0.2 kg·m²
- Same value in lb·ft²4.74608 lb·ft²
Computed by the calculator below using its default values. Change any input to see your own numbers.
I = c·m·r² with the coefficient c set by how far the mass sits from the axis: 2/5 for a solid sphere, 1/2 for a disk, 1 for a hoop. Standard rigid-body results found in any physics text.
What moment of inertia measures
Moment of inertia is rotational mass: how strongly an object resists being spun up or slowed down. It depends not just on how much mass there is, but on where the mass sits. Every kilogram counts by the square of its distance from the axis — mass twice as far out resists four times as much. That squared distance is why all the standard formulas look like a fraction times m·r².
The coefficient encodes the shape. A hoop keeps all its mass at the rim, so c = 1. A solid disk of the same mass and radius hides most of its material near the axis, so c drops to 1/2. A solid sphere tucks mass even closer, giving 2/5. Same mass, same radius, very different effort to spin.
Why the axis matters as much as the shape
A rod spun about its center (I = mL²/12) is four times easier to rotate than the same rod swung from one end (I = mL²/3), because pivoting at the end pushes the far half of the rod out to large distances from the axis. Whenever you move the axis away from the center of mass, inertia grows by the parallel-axis rule I = I_cm + m·d². That is why figure skaters pull their arms in: shrinking r collapses I, and conservation of angular momentum spins them faster.
How it’s calculated
I = c·m·r² (or c·m·L² for rods) with c = 1 (point mass, hoop), 2/5 (solid sphere), 2/3 (thin spherical shell), 1/2 (solid cylinder or disk about its central axis), 1/12 (rod about center), 1/3 (rod about end). Conversions: 1 lb = 0.45359237 kg, 1 in = 0.0254 m, 1 ft = 0.3048 m; 1 kg·m² = 23.7304 lb·ft². These are the standard uniform-density results tabulated in physics references (e.g., Halliday, Resnick & Walker).
Idealized uniform-density shapes about the exact axis listed — real parts with holes, tapers, or offset axes need integration or the parallel-axis theorem (I = I_cm + m·d²).
Standard moments of inertia (m = 1 kg, r or L = 1 m)
| Shape & axis | Formula | I (kg·m²) |
|---|---|---|
| Point mass at radius r | m·r² | 1.000 |
| Hoop / thin-walled cylinder | m·r² | 1.000 |
| Thin spherical shell | (2/3)·m·r² | 0.667 |
| Solid cylinder / disk | (1/2)·m·r² | 0.500 |
| Solid sphere | (2/5)·m·r² | 0.400 |
| Rod, axis through end | (1/3)·m·L² | 0.333 |
| Rod, axis through center | (1/12)·m·L² | 0.083 |
Standard rigid-body results for uniform density about the axes shown; see Halliday, Resnick & Walker, Fundamentals of Physics.
Common mistakes
- Entering the diameter as r — the formulas take radius, and doubling r quadruples I.
- Using the center-axis rod formula for a rod pivoted at its end; the correct value is 4 times larger (1/3 vs 1/12).
- Leaving mass in pounds or length in inches without converting — I scales with m·r², so unit slips compound fast.
- Applying these solid-shape coefficients to hollow or non-uniform parts; a hollow cylinder needs (1/2)m(r₁² + r₂²), not (1/2)mr².
Frequently asked questions
What is the moment of inertia formula?
For a point mass, I = m·r². Extended shapes integrate that over their volume, which collapses to a coefficient times m·r²: (2/5)mr² for a solid sphere, (1/2)mr² for a disk, (1/12)mL² for a rod about its center, and so on.
What units does moment of inertia use?
kg·m² in SI — kilograms times meters squared. Imperial engineering work often uses lb·ft²; 1 kg·m² = 23.7304 lb·ft². This calculator shows both.
Why does a hollow sphere have a larger I than a solid one?
Same mass, but a shell holds all of it at the full radius, while a solid sphere buries most mass near the center. Distance counts squared, so the shell's coefficient (2/3) beats the solid's (2/5).
What if my axis doesn't pass through the center?
Use the parallel-axis theorem: I = I_cm + m·d², where d is the distance from the center-of-mass axis to your axis. The rod-about-its-end formula is exactly this: mL²/12 + m(L/2)² = mL²/3.
Is moment of inertia the same as inertia?
No. Inertia (mass) resists straight-line acceleration; moment of inertia resists angular acceleration and depends on the axis. The same wrench has one mass but a different I for every way you spin it.