Define principal moment of inertia and principal axes

Principal axes

For a plane area, the principal axes are the pair of mutually perpendicular axes through a point (normally the centroid) about which the product of inertia is zero. They are the axes about which the moments of inertia take extreme (maximum and minimum) values. For a section with an axis of symmetry, the axis of symmetry and the axis perpendicular to it are principal axes.

Principal moments of inertia

The moments of inertia about the principal axes are called the principal moments of inertia: the maximum $I_{max}$ about one principal axis and the minimum $I_{min}$ about the other. For a section with $I_x$, $I_y$ and $I_{xy}$ about the centroidal axes:

$$I_{max,min} = \frac{I_x + I_y}{2} \pm \sqrt{\left(\frac{I_x - I_y}{2}\right)^2 + I_{xy}^2}$$

The inclination of the principal axes to the x-axis is given by

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

Two directions, $\theta_p$ and $\theta_p + 90^\circ$, satisfy this equation. Note that $I_{max} + I_{min} = I_x + I_y$ (the polar moment of inertia is constant) and $I_{max}I_{min} = I_xI_y - I_{xy}^2$.