# Perpendicular axis theorem

In physics, the perpendicular axis theorem (or plane figure theorem) can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis perpendicular to the plane, given the moments of inertia of the object about two perpendicular axes lying within the plane. The axes must all pass through a single point in the plane.

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 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.