# Mass Moment of Inertia Calculator

URL: https://speedycalculator.com/calculators/physics/mass-moment-of-inertia/
Last updated: 2026-10-30

> The mass moment of inertia calculator finds the rotational inertia I of 16 common shapes – point masses, rods, disks, cylinders, tubes, spheres, shells, plates, cuboids, cones, tori and ellipsoids – and the radius of gyration.

## Inputs

- Shape
- Mass m (kg, g, mg, t, lb, oz)
- Second mass m₂ (kg, g, mg, t, lb, oz)
- Separation d (m, cm, mm, km, in, ft, yd, mi)
- Radius r (outer / major radius R for the torus) (m, cm, mm, km, in, ft, yd, mi)
- Inner radius (tube radius r for the torus) (m, cm, mm, km, in, ft, yd, mi)
- Length L (m, cm, mm, km, in, ft, yd, mi)
- Side or semi-axis a (m, cm, mm, km, in, ft, yd, mi)
- Side or semi-axis b (m, cm, mm, km, in, ft, yd, mi)

## Results

- Moment of inertia I
- Radius of gyration k = √(I/m)

## Formula

```
Point mass m·r² · Rod (centre) mL²/12 · Disk ½mr² · Hoop mr² · Solid sphere ⅖mr² · Thin shell ⅔mr²
Cuboid m(a² + b²)/12 · Solid cone 3/10·mr² · Torus m(R² + ¾r²) · Ellipsoid m(a² + b²)/5
```

## Explanation

The moment of inertia is the rotational equivalent of mass: it tells you how hard it is to change an object's rotation. It depends not only on mass but on how far that mass is from the axis – a hoop is twice as hard to spin up as a disk of the same mass and radius.

### How to use the Mass Moment of Inertia Calculator

- **Shape**
- **Mass m**
- **Second mass m₂**
- **Separation d**
- **Radius r (outer / major radius R for the torus)**
- **Inner radius (tube radius r for the torus)**
- **Length L**
- **Side or semi-axis a**
- **Side or semi-axis b**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Choose the shape; only its dimensions appear. Axes pass through the centre of mass unless stated (the rod can also be turned about its end).

### Inputs explained

FieldWhat to enterShapeChoose one: Point mass at distance r, Two point masses (separated by d, about the centre of mass), Thin rod about its centre, Thin rod about one end …Mass mUnits: kg, g, mg, t, lb, ozSecond mass m₂Units: kg, g, mg, t, lb, ozSeparation dUnits: m, cm, mm, km, in, ft …Radius r (outer / major radius R for the torus)Units: m, cm, mm, km, in, ft …Inner radius (tube radius r for the torus)Units: m, cm, mm, km, in, ft …Length LUnits: m, cm, mm, km, in, ft …Side or semi-axis aUnits: m, cm, mm, km, in, ft …Side or semi-axis bUnits: m, cm, mm, km, in, ft …

### Example values

How the main result changes when **Mass m** changes and everything else stays at the example values above:

Mass m (kg)Moment of inertia I (kg·m²)10.0451.50.067520.092.50.112530.13540.18

## Worked example

Inputs — Shape: Solid cylinder or disk about its axis; Mass m: 2 kg; Radius r (outer / major radius R for the torus): 0.3 m.

Results — **Moment of inertia I = 0.09 kg·m²**; Radius of gyration k = √(I/m) = 0.21213 m.

## Key terms

- **Moment of inertia**: Resistance to angular acceleration, ∫r² dm.

## FAQ

**Why is a hoop harder to spin than a disk?**
All its mass is at the full radius.

**What is the radius of gyration?**
The distance at which all the mass could be concentrated to give the same I.

**How do I find I about another axis?**
Use the parallel axis theorem: I = I_cm + m·d².

**What are the units?**
kg·m².


## Sources

- [OpenStax – University Physics Volume 1](https://openstax.org/details/books/university-physics-volume-1)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)