Stair Stringer Length

The stringer is the hypotenuse of the whole flight, so $L = \sqrt{H^2 + R_t^2}$ with the total rise and the total run as the legs. An 8 ft rise over a 10 ft run needs a stringer just over 12 ft 9 in on the diagonal, which tells you at once that a 12 ft 2x12 will not do the job and you are buying a 14 footer.

Two practical cautions. This is the line length along the back of the notches, so the actual board must be longer to allow for the plumb cut at the top, the seat cut at the bottom, and the material left below the notches. Framers usually want at least 3$\ frac{1}{2}$ in of solid wood under every notch, or the stringer becomes a row of sawteeth waiting to split. And do not confuse this length with the handrail length, which follows the nosing line and starts and stops in different places.