# Perpendicular axis theorem

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In physics, the perpendicular axis theorem (or plane figure theorem) can be used to determine the second polar moment of area 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 $x$ , $y$ , and $z$ (which meet at origin $O$ ) so that the body lies in the $xy$ plane, and the $z$ 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

$I_{z}=I_{x}+I_{y}$ 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 $I_{x}$ and $I_{y}$ are equal, then the perpendicular axes theorem provides the useful relationship:

$I_{z}=2I_{x}=2I_{y}$ ## Derivation

Working in Cartesian co-ordinates, the moment of inertia of the planar body about the $z$ axis is given by:

$I_{z}=\int \left(x^{2}+y^{2}\right)\,dm=\int x^{2}\,dm+\int y^{2}\,dm=I_{y}+I_{x}$ On the plane, $z=0$ , so these two terms are the moments of inertia about the $x$ and $y$ axes respectively, giving the perpendicular axis theorem. The converse of this theorem is also derived similarly.

Note that $\int x^{2}\,dm=I_{y}\neq I_{x}$ because in $\int r^{2}\,dm$ , 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.