Stand a plank on edge and it carries a floor. Lay the same plank flat and it carries itself, badly. Nothing changed but the direction of the bending, and the property that prices that difference is the second moment of area, $I$ — sometimes called the area moment of inertia, though it has nothing to do with mass.

For a rectangle, $I = \dfrac{b h^{3}}{12}$ — read aloud I equals b h cubed over twelve. Fix the letters before you use them: $b$ is the width, the dimension measured ACROSS the bending, in millimetres; $h$ is the depth, the dimension measured WITH the bending, also in millimetres; and $I$ comes out in mm⁴, a fourth power of length, which is a unit worth pausing over. It is not an area and it is not a volume. It is a measure of how far the material sits from the axis it is being bent about, and the answer counts distance twice over.

That cube is the whole lesson. Double the depth and $I$ goes up eightfold; double the width and it merely doubles. So a 50 × 200 joist on edge is sixteen times stiffer than the same joist lying flat — and the exam's favourite mistake is writing $b^{3}h$ instead of $bh^{3}$. Say the formula out loud each time and the cube lands on the right letter.

A solid round bar gets its own line: $I = \dfrac{\pi d^{4}}{64}$, where $d$ is the diameter in millimetres. Watch that word. Feed a radius into a formula written for a diameter and the fourth power turns a factor of two into a factor of sixteen. And do not reach for $\dfrac{\pi d^{4}}{32}$ — the /32 form is $J$, the polar second moment, which answers twisting. For a circle $J$ is exactly twice $I$, and the two are not interchangeable.