The formula
d = a√2
Where it comes from
The diagonal cuts the square into two right triangles with equal legs. Pythagoras gives d² = a² + a² = 2a², so the diagonal is the side times the square root of two.
How to work it out by hand
- Square the side and double it
- Take the square root
- Equivalently, just multiply the side by 1.41421356
- To go the other way, divide the diagonal by √2
What is worth knowing
This is the ratio that broke Greek mathematics. The Pythagoreans held that every quantity was a ratio of whole numbers, and the diagonal of a unit square is √2, which is not — the proof is short and devastating, and the discovery is said to have been kept secret. It is also the ratio behind A-series paper: an A4 sheet halved gives two A5 sheets of exactly the same proportions, which only works if the sides are in the ratio 1:√2. That is why photocopiers scale A4 to A3 with a single 141% button.