Moment of Inertia Calculator: Cylinder, Sphere, Rod, Hoop and Plate, with the Parallel Axis Theorem

Rotational inertia of the shapes you meet most.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose the shape and enter the mass and the dimensions.

Enter the distance to a parallel axis if you want one, and the rotation speed.

The calculator gives the moment of inertia, the radius of gyration, the parallel axis value, the rotational energy and the angular momentum.

Formulas used

  • I = \tfrac12mr^2\ (\text{disc})
  • I = \tfrac25mr^2\ (\text{sphere})
  • I_p = I + md^2
  • E_k = \tfrac12I\omega^2

Worked example

With the default numbers in the calculator above:

  1. Moment of inertia of the solid cylinder or disc: 0.2 kg m2
  2. Radius of gyration (the radius at which all the mass could be placed to have the same moment of inertia): 0.141421 m
  3. Rotational kinetic energy at that speed: 394.784 J (at 62.8319 rad/s)
  4. Angular momentum at that speed: 12.5664 kg m2/s
  5. Moment of inertia about a parallel axis 0.5 m from the centre axis (parallel axis theorem): 2.7 kg m2
  6. Rotational kinetic energy about that axis at the same speed: 5329.59 J

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Shape Solid sphere, about a diameter.

  1. Moment of inertia of the solid sphere: 0.16 kg m2
  2. Radius of gyration (the radius at which all the mass could be placed to have the same moment of inertia): 0.126491 m
  3. Rotational kinetic energy at that speed: 315.827 J (at 62.8319 rad/s)
  4. Angular momentum at that speed: 10.0531 kg m2/s
  5. Moment of inertia about a parallel axis 0.5 m from the centre axis (parallel axis theorem): 2.66 kg m2
  6. Rotational kinetic energy about that axis at the same speed: 5250.63 J

Common questions

What is the moment of inertia?

The rotational equivalent of mass. It says how hard it is to change the rotation of an object, and it depends on how far the mass is from the axis.

What is the parallel axis theorem?

The moment of inertia about any axis is the moment about a parallel axis through the centre plus the mass times the distance squared.

Why is a hoop harder to spin than a disc?

For the same mass and radius, all the mass of a hoop is at the rim, so its moment of inertia is twice that of a disc.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any physics or engineering mechanics text (moments of inertia).

Need a calculator we do not have?

Tell us what you would like to calculate. We read every request and build the most asked-for ones.

We store your request (and your email if you give it) to decide what to build. We do not store the numbers you type into calculators.

Embed this calculator on your own page

Copy this code into your page. The calculator runs inside the frame and links back to Drafting Central.

Need this drawn, designed or checked?

Drafting Central is a mechanical engineering and drafting business working with customers across Australia and around the world. A calculator gives you the numbers; we can turn them into drawings and designs.

Need help with a real project?

Talk to Drafting Central →

← All engineering calculators