The formulas
With total rise H, maximum riser hmax and going g:
- Risers: R = ⌈H ÷ hmax⌉. Round up; rounding down makes every riser too tall.
- Riser height: h = H ÷ R. All risers are equal.
- Treads: T = R − 1 when the upper floor is the top step, which is the usual case. T = R when a tread is built level with the upper floor.
- Total run: run = T × g, from the face of the bottom riser to the face of the top riser.
- Angle: θ = atan(h ÷ g).
Two more are useful: the pitch-line length R × √(h² + g²), a geometric reference along the nosings, and 2h + g, a centuries-old rule of thumb for how a step feels. Neither is a cut length or a code requirement.
Worked example in inches
H = 108 in (9 ft), hmax = 7 3/4 in, g = 10 1/2 in, upper floor as the top step.
- R = ⌈108 ÷ 7.75⌉ = ⌈13.935⌉ = 14
- h = 108 ÷ 14 = 7.7143 in ≈ 7 11/16 in
- T = 14 − 1 = 13
- run = 13 × 10.5 = 136.5 in = 11 ft 4 1/2 in
- θ = atan(7.7143 ÷ 10.5) = 36.3°
Pitch-line length: 14 × √(7.7143² + 10.5²) = 14 × 13.029 = 182.41 in. 2h + g = 25.93 in.
Worked example in millimetres
H = 2600 mm, hmax = 190 mm, g = 250 mm.
- R = ⌈2600 ÷ 190⌉ = ⌈13.68⌉ = 14
- h = 2600 ÷ 14 = 185.7 mm
- T = 13
- run = 13 × 250 = 3250 mm (3.250 m)
- θ = atan(185.7 ÷ 250) = 36.6°
The 190 mm maximum here is just an example input, not a rule for any country. Use the limit that applies where you build.
What one more riser changes
Adding a riser makes every step lower and the whole stair one going longer. With the same 108 in rise and 10 1/2 in going:
| Risers | Riser height | Treads | Total run | Angle | Under 7 3/4 in? |
|---|---|---|---|---|---|
| 13 | 8 5/16 in | 12 | 10 ft 6 in | 38.4° | No, too tall |
| 14 | 7 11/16 in | 13 | 11 ft 4 1/2 in | 36.3° | Yes |
| 15 | 7 3/16 in | 14 | 12 ft 3 in | 34.4° | Yes |
| 16 | 6 3/4 in | 15 | 13 ft 1 1/2 in | 32.7° | Yes |
| 17 | 6 3/8 in | 16 | 14 ft 0 in | 31.2° | Yes |
Thirteen risers would be 8 5/16 in, over the maximum, so 14 is the steepest choice. Fifteen or sixteen give a gentler stair if the floor space allows.
Working backwards from the space you have
If the run must fit a fixed length L, the going is limited to L ÷ T. For the 14-riser stair above, a 130 in space allows at most 130 ÷ 13 = 10 in per going: the IRC's minimum tread depth, with the steepest result, atan(7.714 ÷ 10) = 37.6°. The planned 10 1/2 in going needs 11 ft 4 1/2 in, which is 6 1/2 in more than the space. In that situation the usual options are a shallower going (if your limits allow), a turn with a landing, or more floor space. Fewer risers would make each riser taller than your maximum.
Two conventions that change the answer
- Top of the stair. If a tread sits level with the upper floor (a flush-mounted stringer, common on decks), T = R, so the same 14-riser stair has 14 treads and runs 147 in instead of 136.5 in.
- Going vs tread depth. Run uses the going, the nosing-to-nosing distance. A tread board with a 1 in nosing is 1 in deeper than the going. Using the board depth overstates the run by T inches.
Rounding without drift
7.7143 in is not exactly 7 11/16 in (7.6875 in). Fourteen risers at 7 11/16 in would add up to 107.625 in, 3/8 in short of the floor. Keep the exact value in the calculation, round only to mark, and mark each riser from a single reference point rather than stepping off one from the last. The stringer calculator lists every mark measured from the first.