A worked example: 3m rise, 4m run
A staircase with a 3m total rise and a 4m total run needs a stringer 5m long, at an angle of 36.87° — the classic 3-4-5 right triangle, giving a perfectly whole-number result.
The formula: rise, run, and the Pythagorean theorem
Stringer length = √(rise² + run²). The stringer is the diagonal board connecting the bottom of the stairs to the top, so treating the rise and run as the two legs of a right triangle and finding the hypotenuse gives exactly the length needed.
A steeper staircase: 3m rise, 2.5m run
Keeping the same 3m rise but shortening the run to 2.5m needs a shorter 3.9051m stringer, but at a much steeper 50.19° angle — less horizontal run for the same vertical climb means a steeper staircase, even though the stringer itself is shorter.
Why the angle formula is arctangent, not arcsine or arccosine
Angle = arctan(rise ÷ run) — the ratio of the two legs directly opposite and adjacent to the stair's angle from horizontal. This is the standard way to recover an angle from a right triangle's two legs, without needing the hypotenuse (stringer length) at all.
What this raw length doesn't include
This is the raw diagonal distance only — an actual stringer board needs extra length beyond this for the notched step cuts at each tread and the connections at the top and bottom, which a carpenter adds on top of this baseline figure.