${\mathrm{I}}_{\mathrm{x}\text{'}}=\frac{{\mathrm{I}}_{\mathrm{x}}+{\mathrm{I}}_{\mathrm{y}} }{2}+\frac{{\mathrm{I}}_{\mathrm{x}}-{\mathrm{I}}_{\mathrm{y}} }{2}\cos 2\mathrm{\theta } - {\mathrm{I}}_{\mathrm{xy}}\sin 2\mathrm{\theta }$

${\mathrm{I}}_{\mathrm{y}\text{'}}=\frac{{\mathrm{I}}_{\mathrm{x}}+{\mathrm{I}}_{\mathrm{y}} }{2}-\frac{{\mathrm{I}}_{\mathrm{x}}-{\mathrm{I}}_{\mathrm{y}} }{2}\cos 2\mathrm{\theta } + {\mathrm{I}}_{\mathrm{xy}}\sin 2\mathrm{\theta }$

${\mathrm{I}}_{\mathrm{x}\text{'}\mathrm{y}\text{'}}=\frac{{\mathrm{I}}_{\mathrm{x}}-{\mathrm{I}}_{\mathrm{y}} }{2}\sin 2\mathrm{\theta } + {\mathrm{I}}_{\mathrm{xy}}\cos 2\mathrm{\theta }$

There exists one set of axes for which the moments of inertia of an area are maximum and minimum. These axes are called the principal axes of an area. The values of the maximum and minimum moments of inertia can be found from:

${\mathrm{I}}_{p1},{\mathrm{I}}_{p2}=\frac{{\mathrm{I}}_x+{\mathrm{I}}_y}{2}\pm \sqrt{{\left(\frac{{\mathrm{I}}_x-{\mathrm{I}}_y}{2}\right)}^2+{I_{xy}}^2}$

Relative to the $\mathrm{x}$ and $\mathrm{y}$ axes, the principal axes are oriented at an angle given by:

$\tan 2{\theta }_p= - \frac{2I_{xy}}{I_x-I_y}$