Radius of Gyration (r = √(I/A))

The radius of gyration answers a neat question: if you scraped all the material of a section into two thin strips, how far from the axis would they have to sit to give the same I? That distance is $r = \sqrt{I/A}$, and it is the honest measure of how spread out a section is, independent of how much metal it contains. For a rectangle bending about its strong axis $r = h/\sqrt{12} = 0.2887h$, so a 50 × 100 mm bar has r = 0.02887 m about that axis — and only 0.01443 m about the weak one.

Its whole purpose is column design, where the slenderness ratio KL/r decides everything. Because a column buckles about whichever axis has the smaller r, the least radius of gyration is the one that matters — a wide-flange section is often several times weaker about its y-axis than its x-axis, which is why columns get their weak axis braced. Handbook tables list $r_x$ and $r_y$ for every rolled shape precisely so this ratio can be formed without touching I at all. Remember I goes in as a plain number in m⁴ while A uses a real area unit, so the answer comes out as a length.