Perpendicular Axis Theorem

1 min read Last updated Sat Jun 27 2026 08:46:52 GMT+0000 (Coordinated Universal Time)

Applies only to flat (planar) objects lying in the $xy$ -plane. For any 2 perpendicular axes $x$ , $y$ in the plane of the object and a third axis $z$ perpendicular to the plane:

I_{zz} = I_{xx} + I_{yy}

The same holds for any other pair of in-plane perpendicular axes $m$ , $n$ sharing the same $z$ :

I_{zz} = I_{mm} + I_{nn}

If $I_{xx}$ is at maximum, $I_{yy}$ is at minimum (their sum is fixed).

Derivation

For a mass element $dm$ at position $(x, y)$ :

I_{zz} = \int r^2 \,\text{d}m = \int (x^2 + y^2)\,\text{d}m = \int y^2\,\text{d}m + \int x^2\,\text{d}m = I_{xx} + I_{yy}

Example

Thin disk of radius $R$ and mass $M$ : $I_{xx} = I_{yy} = \dfrac{MR^2}{4}$ by symmetry.

I_{zz} = \frac{MR^2}{4} + \frac{MR^2}{4} = \frac{MR^2}{2}

Was this helpful?