Example 1
m = 2 kg, r = 0.5 m → I = 0.5.
Moment of inertia (point mass) calculator
Type m [kg] and r [m]. The calculator computes I = m r². 2 kg and 0.5 m is 0.5 kg·m². The same 2 kg at r = 0.1 m gives 0.02. Radius squared bites harder than mass.
Angular momentum: L = I·ω. Energy: Eₖ = ½Iω². Torque: M = F·r·sinα.
Result
Mass m and Distance from axis r. The result shows up here.
How it works
Point-mass (or thin-hoop) inertia is I = m r². At 2 kg and 0.5 m: I = 2 × 0.25 = 0.5 kg·m². The same 2 kg at r = 0.1 m gives 0.02 kg·m². At 1 kg and 2 m you get 4 kg·m². Radius squared bites harder than mass: doubling r from 0.5 m to 1 m raises I from 0.5 to 2 kg·m².
The form has two fields: m in kilograms and r in meters. The result is in kg·m². This is not the sphere formula. A solid sphere about its center is ⅖ m r²; a cylinder has other factors. Here a point or a hoop. A comma and a period are the same r: 0,5 and 0.5.
m must be positive. r = 0 gives I = 0: mass on the axis adds no rotational inertia. Typed 0 in m also gives I = 0. Both fields need a number before 0.5 kg·m² can appear.
With this I you go to L = Iω or to Eₖ = ½Iω². This calculator stops at I. Torque M = F r sinα is twist, not I.
If you know I and want r at a known m, invert it yourself: r = √(I/m). Here you only go one way.
Type 2 and 0.5, click Calculate, and match 0.5 kg·m². The header symbol does not weigh a kilogram. Treat I as resistance to a change of spin, not as energy.
How to use
- In the first field enter mass m in kilograms, for example 2.
- In the second field enter r in meters, for example 0.5. From the axis to the point.
- Click Calculate. The calculator computes m r². 2 kg and 0.5 m give I = 0.5 kg·m². At r = 0.1 m that is 0.02.
- r = 0 gives I = 0: mass on the axis. Both fields need a number. This is not the sphere formula.
- With this I, open L = Iω or Eₖ = ½Iω². Torque is a separate card.
Formula
I = m·r2
Unit: kg·m².
I, m, and r for a point mass
Point mass: I = m·r². 2 kg and 0.5 m is 0.5 kg·m². The same 2 kg at r = 0.1 m gives 0.02. Radius squared bites harder than mass.
- I
- Moment of inertia [kg·m²]. 2 × 0.25 = 0.5. Input to L = Iω on the next card.
- m
- Point mass [kg]. 2 kg. Linear, not squared.
- r
- Distance from the axis [m]. 0.5 m. r²: twice as close, four times smaller I.
Real-life examples
Example 2
m = 2 kg, r = 0.1 m.
Example 3
m = 1 kg, r = 2 m.
Example 4
m = 0.2 kg, r = 0.8 m.
Example 5
m = 5 kg, r = 0.25 m.
Example 6
m = 3 kg, r = 1 m vs 0.5: I ×4.
Example 7
m = 0.05 kg, r = 0.3 m.
Example 8
m = 10 kg, r = 0.2 m.
Example 9
m = 4 kg, r = 0.4 m.
Ways to use this calculator
- You compute 0.5 kg·m² from 2 kg at r = 0.5 m.
- You feed that I into L = Iω or into ½Iω².
Frequently asked questions
How much I at 2 kg and r = 0.5 m?
Moment of inertia is 0.5 kg·m². That is 2 × 0.25. At r = 0.1 m the same 2 kg give only 0.02 kg·m².
Which units do I type?
Mass m [kg], radius r [m]. Result I [kg·m²]. This is a point mass or a hoop, not a ⅖ m r² sphere.
Does r = 0 m give I = 0?
Yes. Mass on the axis gives no rotational inertia. The product m r² vanishes when the radius is zero.
Does this work for a sphere?
No. A sphere about its center is ⅖ m r². Here you have a point mass or a thin hoop, the full m r².
Where is L = Iω?
On the angular-momentum page. There you multiply this I by ω. Here you only compute I = m r².
Does a comma in 0.5 work?
Yes. 0.5 and 0,5 are the same r [m]. A comma and a period mean the same value.
How much at 1 kg and r = 2 m?
You get 4 kg·m². One kilogram at two metres: 1 × 4.
How does I grow with r at 2 kg?
With the square of the radius. Twice r gives four times I. At 2 kg and 1 m that is 2 kg·m², not 1.
Where is rotational energy?
On the rotational-kinetic-energy page when it is on the hub. Here only I, no ½ I ω².
Is this the same as torque?
No. Torque is M = F r, in newton-metres. Here I says how the mass resists being spun about the axis.
Knowledge sources
The formula is the school one. Units follow SI; NIST SP 330 and BIPM define the measures, not your result.
Page updated in 2026.