Rectangular Cross-Section

For a rectangular cross-section of height $h$ and width $w$, area $S=hw$, the moment of inertia, per unit mass, about the horizontal midline is given by

$$I_w = w\int_{-h/2}^{h/2} y^2 dy = w\left.\frac{1}{3}y^3\right\vert _{-h/2}^{h/2} = \frac{Sh^2}{12}$$

The radius of gyration about this axis is then

$$r_w = \sqrt{\frac{I_w}{S}} = \sqrt{\frac{h^2}{12}} = \frac{h}{2\sqrt{3}}.$$

Similarly, the radius of gyration about a vertical axis passing through the center of the cross-section is $r_h=w/(2\sqrt{3})$.

The radius of gyration can be thought of as the ``effective radius'' of the mass distribution with respect to its inertial response to rotation (``gyration'') about the chosen axis.

Most cross-sectional shapes (e.g., rectangular), have at least two radii of gyration. A circular cross-section has only one, and its radius of gyration is equal to half its radius, as shown in the next section.