Moment of Inertia Calculator
Moment of inertia of a cylinder, disc, ring, tube, sphere, shell, rod, plate or point mass about its standard axis, with the parallel-axis theorem for a shifted axis and the rotational kinetic energy ½ I ω².
Result…
How it's calculated
Solid cylinder or disc: I = ½ m R² Thin ring or thin-walled cylinder: I = m R² Tube: I = ½ m (R_inner² + R_outer²) Solid sphere: I = ⅖ m R²; thin shell: I = ⅔ m R² Rod: I = ¹⁄₁₂ m L² (centre), ⅓ m L² (end) Plate: I = ¹⁄₁₂ m (a² + b²) Parallel axis: I = I_cm + m d² KE = ½ I ω²; L = I ω
Moment of inertia is rotational inertia: how hard an object is to spin up or slow down about an axis. It depends on the mass and on how far that mass sits from the axis, so a ring is harder to spin than a disc of the same mass and radius. The shape formulas are the standard ones for uniform bodies, as tabled by HyperPhysics (Georgia State University). The parallel-axis theorem adds m d² for an axis moved a distance d from the centre of mass. Rotational kinetic energy is ½ I ω² with ω in rad/s. Unit factors are exact: 1 lb = 0.45359237 kg, 1 ft = 0.3048 m, so 1 lb·ft² = 0.0421401100938048 kg·m².
Your numbers, step by step
About the Moment of Inertia calculator
This moment of inertia calculator gives the rotational inertia of common uniform shapes about their standard axes: a solid cylinder or disc, a thin ring or thin-walled cylinder, a thick-walled tube, a solid sphere, a thin spherical shell, a rod about its centre or end, a rectangular plate and a point mass. Turn on the parallel-axis option to move the axis a distance d from the centre of mass, I = I_cm + m d². Add an angular speed in rad/s or rpm to get the rotational kinetic energy ½ I ω² and the angular momentum. Mass works in kg, g or lb and lengths in metres, centimetres, millimetres, feet or inches, with the answer in kg·m² and lb·ft². It suits flywheels, pulleys, rotors and physics problems, and every step is shown with your numbers.
How to use it
- Pick the “Shape and axis” that matches your object.
- Enter the “Mass” and the sizes the shape needs, such as “Radius”, “Rod length” or “Plate length” and “Plate width”, and choose the units.
- To use an axis that does not pass through the centre, set “Axis” to the shifted option and enter the “Distance between the axes”.
- Enter an “Angular speed” for the kinetic energy, or 0 to skip it, and read the moment of inertia, radius of gyration and energy.
Worked example
A 20 kg flywheel disc of radius 0.2 m has I = ½ × 20 × 0.2² = 0.4 kg·m², about 9.49 lb·ft². Spinning at 3000 rpm (314.16 rad/s), it stores ½ × 0.4 × 314.16² = 19,739 J of kinetic energy, about 14,559 ft·lbf, and has an angular momentum of 125.7 kg·m²/s. Its radius of gyration is 0.141 m.
Updated
Questions people ask
How do I calculate the moment of inertia of a disc?
For a uniform solid disc or cylinder about its central axis, I = ½ m R². Mass in kg and radius in metres give kg·m². A thin ring of the same mass and radius has twice as much, m R², because all its mass is at the rim.
What is the parallel-axis theorem?
It moves the axis sideways: I = I_cm + m d², where I_cm is about a parallel axis through the centre of mass and d is the distance between the axes. A rod of length L about its centre has ¹⁄₁₂ m L²; shifting by L ÷ 2 gives ⅓ m L², the value about its end.
What is the formula for rotational kinetic energy?
KE = ½ I ω², with ω in rad/s. Convert rpm first with ω = 2π × rpm ÷ 60; putting rpm straight into the formula is a common mistake that gives a far too large answer.
Where do these shape formulas come from?
They are the standard results for uniform bodies, as tabled by HyperPhysics at Georgia State University, with the tube and plate formulas as given in the list of moments of inertia.
What are the limits of this moment of inertia calculator?
It assumes an evenly dense object about the axis named. Real parts with holes, hubs, spokes or uneven walls differ, so add up the pieces or use a CAD model, and get a qualified engineer to check anything that spins fast enough to be dangerous. For straight-line kinetic energy, use the kinetic energy calculator.