Square Diagonal

Cut a square corner to corner and you get two right triangles whose legs are both $s$, so Pythagoras gives $d = s\sqrt{2} \approx 1.414s$. Carpenters use this constantly in reverse: to check that a frame is square, measure both diagonals — if they match, the corners are true. It also sets the largest square that fits through a doorway, and the reach of a diagonal brace.

The number $\sqrt{2}$ itself has a claim to fame: it was the first quantity proved irrational, around the 5th century BC, precisely because it is the diagonal of the unit square. Paper sizes exploit it too — A4 and its siblings have a $\sqrt{2}$ aspect ratio, which is why folding one in half yields the same proportions again.