Every compression member has two ways to fail, and they are not the same failure. A stubby one crushes — it simply runs out of strength. A long one buckles — it bows sideways while the material is still relaxed, and the load it lets go at has almost nothing to do with how strong the steel is. Which one you get is not a matter of opinion. One number decides it, and this lesson builds that number in two moves.

First the shape. $r = \sqrt{\dfrac{I}{A}}$ — read aloud: r equals the square root of I over A. Here $r$ is the radius of gyration, a length in millimetres; $I$ is the area moment of inertia of the cross-section, in $\mathrm{mm^4}$, the shape's resistance to bending; and $A$ is the cross-sectional area, in $\mathrm{mm^2}$. Note that r is not a radius you could measure with calipers — it is the distance at which all that area COULD sit and still bend the same way. A column always goes over about its weakest axis, so it is the SMALLEST r of the section that counts.

Then the column. $\lambda = \dfrac{KL}{r}$ — lambda equals K L over r, and $\lambda$ is the Greek letter lambda. $\lambda$ is the slenderness ratio, a naked number with no units at all, because it is a length over a length. $L$ is the unbraced length between supports; $K$ is the effective length factor that reads the end conditions (1.0 pinned at both ends, 0.5 fixed at both, 2.0 fixed at the base and free at the top); and $r$ is the radius of gyration you just found. Because λ is a ratio of two lengths, both must arrive in the SAME unit — a length in metres divided by a radius in millimetres is wrong by a thousand, and it is the single most common slip in column work.