# Perpendicular axis theorem

The perpendicular axis theorem states that the moment of inertia of a plane lamina about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia of the lamina about the two axes at right angles to each other, in its own plane intersecting each other at the point where the perpendicular axis passes through it.

Define perpendicular axes ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ (which meet at origin ${\displaystyle O}$) so that the body lies in the ${\displaystyle xy}$ plane, and the ${\displaystyle z}$ axis is perpendicular to the plane of the body. Let Ix, Iy and Iz be moments of inertia about axis x, y, z respectively, the perpendicular axis theorem states that[1]

${\displaystyle I_{z}=I_{x}+I_{y}}$

This rule can be applied with the parallel axis theorem and the stretch rule to find polar moments of inertia for a variety of shapes.

If a planar object (or prism, by the stretch rule) has rotational symmetry such that ${\displaystyle I_{x}}$ and ${\displaystyle I_{y}}$ are equal, then the perpendicular axes theorem provides the useful relationship:

${\displaystyle I_{z}=2I_{x}=2I_{y}}$

## Derivation

Working in Cartesian co-ordinates, the moment of inertia of the planar body about the ${\displaystyle z}$ axis is given by:[2]

${\displaystyle I_{z}=\int \left(x^{2}+y^{2}\right)\,dm=\int x^{2}\,dm+\int y^{2}\,dm=I_{y}+I_{x}}$

On the plane, ${\displaystyle z=0}$, so these two terms are the moments of inertia about the ${\displaystyle x}$ and ${\displaystyle y}$ axes respectively, giving the perpendicular axis theorem. The converse of this theorem is also derived similarly.

Note that ${\displaystyle \int x^{2}\,dm=I_{y}\neq I_{x}}$ because in ${\displaystyle \int r^{2}\,dm}$, r measures the distance from the axis of rotation, so for a y-axis rotation, deviation distance from the axis of rotation of a point is equal to its x co-ordinate.

## References

1. Paul A. Tipler (1976). "Ch. 12: Rotation of a Rigid Body about a Fixed Axis". Physics. Worth Publishers Inc. ISBN 0-87901-041-X.
2. K. F. Riley, M. P. Hobson & S. J. Bence (2006). "Ch. 6: Multiple Integrals". Mathematical Methods for Physics and Engineering. Cambridge University Press. ISBN 978-0-521-67971-8.