Perpendicular axis theorem

The perpendicular axis theorem (or plane figure theorem) states that, "The moment of inertia (I_z) of a laminar body about an axis(𝘻)

) perpendicular to its plane is the sum of its moments of inertia about two mutually perpendicular axes (x and y) in its plane, all the three axes being concurrent. "

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 I_x, I_y and I_z be moments of inertia about axis x, y, z respectively. Then 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 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 coordinates, the moment of inertia of the planar body about the $z$ axis is given by:

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

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 coordinate.

See also

• Parallel axis theorem
• Stretch rule