About the Moment of Inertia Calculator
This moment of inertia calculator gives the mass moment of inertia – the rotational equivalent of mass – for the shapes that come up most in physics and engineering: solid cylinders and disks, thin hoops, thick-walled tubes, solid and hollow spheres, thin rods, rectangular plates and point masses. Enter the mass and dimensions, and optionally an offset to move the axis using the parallel axis theorem.
Besides the moment of inertia about your chosen axis, it reports the value about the centre of mass, the radius of gyration, and – if you give a rotation speed in rpm – the rotational kinetic energy and angular momentum. It suits students solving rolling and rotation problems, engineers sizing flywheels, motors and shafts, and hobbyists building robots, drones or spinning tops. The comparison chart shows how shape alone changes the inertia for the same mass and radius.
All formulas assume uniform density and rigid bodies, and "thin" shapes assume the wall or rod thickness is small compared with the other dimensions. This is mass moment of inertia (kg·m²), not the area moment of inertia (m⁴) used for beam bending.
With the default inputs, the moment of inertia about your axis is 0.45 kg·m². Change any value above to recalculate instantly.
How to use the moment of inertia calculator
- 1Choose the shape and axis that match your object.
- 2Enter the mass and pick the mass and length units.
- 3Enter the dimensions shown for that shape.
- 4Add an axis offset d if the rotation axis does not pass through the centre.
- 5Optionally enter a speed in rpm to get spin energy and angular momentum.
Formula and method
The moment of inertia I = Σm·r² measures how mass is spread around an axis: the farther the mass sits from the axis, the harder the body is to spin up or slow down. Integrating over each uniform shape gives the standard results – a hoop, with all mass at radius R, has the maximum MR², while a solid disk has ½MR² and a solid sphere ⅖MR².
The parallel axis theorem moves the axis: about any axis parallel to one through the centre of mass and a distance d away, I = I_cm + Md². That is how a rod about its end gives ¹⁄₁₂ML² + M(L/2)² = ⅓ML². With angular speed ω = 2π·rpm/60, rotational kinetic energy is ½Iω², angular momentum is Iω and the radius of gyration is k = √(I/M).
- I
- Moment of inertia about the chosen axis (kg·m²)
- I_cm
- Moment of inertia about the parallel axis through the centre of mass
- M
- Mass (kg)
- R, r
- Outer and inner radius (m)
- L, a, b
- Rod/cylinder length and plate sides (m)
- d
- Axis offset from the centre of mass (m)
Worked examples
10 kg flywheel disk, 0.3 m radius, at 300 rpm
I = ½ × 10 × 0.3² = 0.45 kg·m². At 300 rpm, ω = 31.42 rad/s, so KE = ½ × 0.45 × 31.42² = 222 J. The radius of gyration is √(0.45/10) = 0.212 m.
2 kg, 1 m rod swung about one end
About its centre, I = ¹⁄₁₂ × 2 × 1² = 0.167 kg·m². Shifting the axis 0.5 m to the end adds 2 × 0.5² = 0.5, giving 0.667 kg·m² – the textbook ⅓ML².
5 kg bowling-type solid sphere, 10 cm radius
10 cm = 0.1 m, so I = ⅖ × 5 × 0.1² = 0.02 kg·m².
3 kg steel pipe, 50 mm outer and 40 mm inner radius
I = ½ × 3 × (0.05² + 0.04²) = ½ × 3 × 0.0041 = 0.00615 kg·m², between a solid bar (0.00375) and a thin tube (0.0075) of the same mass.
0.5 kg point mass on a 1.2 m arm
For a point mass all the inertia comes from its distance: I = Md² = 0.5 × 1.2² = 0.72 kg·m².
Frequently asked questions
What is the moment of inertia?+
It is a body's resistance to changes in its rotation about an axis, the rotational counterpart of mass. It depends on both the mass and how far that mass is from the axis: I = Σ m r², measured in kg·m².
What is the parallel axis theorem?+
If you know the moment of inertia about an axis through the centre of mass, the value about any parallel axis a distance d away is I = I_cm + M d². It always increases the inertia, so the centre-of-mass axis gives the minimum.
Why does a hollow cylinder roll slower than a solid one?+
A hollow cylinder (I = MR²) keeps all its mass at the rim, so more of its energy goes into spinning and less into moving forward. A solid cylinder (½MR²) accelerates faster down a slope, and a solid sphere (⅖MR²) faster still.
What is the radius of gyration?+
The radius of gyration k = √(I/M) is the distance from the axis at which all the mass could be concentrated to give the same moment of inertia. For a solid disk it is R/√2 ≈ 0.707R.
Is this the same as the area moment of inertia for beams?+
No. Mass moment of inertia (kg·m²) governs rotation. The area or second moment of area (m⁴ or in⁴) describes a cross-section's resistance to bending and is used in beam deflection and stress calculations.