Engineering Calc Tools

Dynamics

Mass Moment of Inertia Calculator

Moment of inertia and radius of gyration of a body built from rods, disks, spheres and plates, about any parallel axis.

Tool no.
ECT-D09
Rev
1.0
Updated
Oct 2026
Method
Parallel-axis theorem
Examples

Given

rod m= L= d= · rodend m= L= · disk m= r= d= (about its own axis) · diskd m= r= h= (about a diameter) · ring m= r= · tube m= ro= ri= · sphere m= r= · shell m= r= · plate m= a= b= · plateedge m= a= · cone m= r= · point m= d=. d is the distance from the part’s own parallel axis to the axis of rotation. Put hole in front to subtract.

Answers

Moment of inertia =—
Runs in your browser. Nothing you enter is uploaded.Parallel-axis theorem
How to use this calculator

List each part with its mass, dimensions, and the distance d from its own center (its own parallel axis) to the axis you want. Every part’s axis must be parallel to that axis.

Formulas

Parallel axisI = IG + md²
Radius of gyrationk = √(I/m)
Common shapesrod mL²/12, disk ½mr², sphere ⅖mr², ring mr²

Worked example

A 3 kg, 0.8 m rod pinned at one end with a 5 kg, 0.1 m disk centred 0.9 m from the pin (the default)

  1. Rod: IG = 3(0.8)²/12 = 0.16, plus 3(0.4)² = 0.48: 0.64 kg·m².
  2. Disk: ½(5)(0.1)² = 0.025, plus 5(0.9)² = 4.05: 4.075 kg·m².
  3. IO = 4.715 kg·m², m = 8 kg, k = 0.768 m.

Frequently asked questions

What is d for a rod pinned at its end?

Use rod with d = L/2, or rodend directly; both give mL²/3.

Can I use weights instead of masses?

Divide weight by g first. In US units, mass is in slugs: m = W/32.2.