Laying out a set of stairs
stair mathsrise and runstringer layout
Riser count, the 2R plus T comfort rule, total run and stringer length — the full sequence for cutting stairs.
Stair Riser Count from Total Rise
Divides the total rise, finished floor to finished floor, by the target riser height to give the number of risers a stair needs.
Stair Comfort Rule (2R + T)
The stride rule that keeps a stair comfortable: twice the riser plus one tread should land near 25 inches, or 630 mm in metric practice.
Stair Total Run from Risers and Tread Depth
The floor space a straight flight consumes: one fewer tread than there are risers, because the upper floor serves as the last tread.
Stair Stringer Length
The straight-line length of the stringer board, the hypotenuse of the total rise and the total run.
How they fit together
Stairs are laid out in a fixed order and each formula feeds the next. Start with the total rise, floor to floor, and divide by a target riser height to get the number of risers. That answer almost never lands on a whole number, so you round and then divide back to get the actual riser height, which is why the count comes first and the height second.
The 2R + T rule is the comfort check, and it is older than any building code: two risers plus one tread should land near 25 inches, or about 630 mm. It encodes how a human stride shortens as it climbs. Stairs that violate it feel wrong underfoot even when every dimension is legal.
Total run then tells you whether the staircase fits the room, and the stringer length is the diagonal you actually cut, which is simply Pythagoras on the rise and the run. Codes govern the limits on riser and tread, and they vary by jurisdiction and by occupancy, so check locally before cutting.