Warm a solid and it grows — every metre of it, by a tiny fixed fraction, for every degree. The formula is $\Delta L = \alpha L_0 \Delta T$ — read aloud: delta-L equals alpha L-nought delta-T. $\Delta L$ is the growth in length — the millimetres you are solving for. Two new sounds join it: $\alpha$ is the Greek letter alpha, the material's expansion fraction per degree, and $L_0$ is said L-nought — the length in metres before the warm-up, with $\Delta T$ the degrees it warms by. For steel, $\alpha = 12 \times 10^{-6}\ /^{\circ}\mathrm{C}$: twelve millionths per degree.

Twelve millionths sounds like nothing, and for a ruler on your desk it is. But feed $L_0$ a fifty-metre rail and $\Delta T$ a Canadian summer, and a tiny fraction of a very long thing becomes real millimetres — which is exactly why rails are laid with gaps and bridge decks ride on sliding joints. So calibrate your expectations before computing: the right answer lives in millimetres. If yours comes out in metres, a ×1000 crossing went missing somewhere — and that is this lesson's favourite trap.