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Woodworking Math

Stair calculator

Enter the total rise from floor to floor. You get the number of risers and their height, the treads and total run, the stair's angle, the stringer length and the framing-square settings for cutting it, each checked against the IRC stair limits.

Finished floor to finished floor: measure where the stair will land.
Nosing to nosing. 10 in is the IRC’s least.
Enter it if the stair must fit a space: the treads share it.
More: riser limit, tread thickness, headroom, stringer
7 3/4 in is the IRC’s most; a lower one adds risers.
Leave blank for the fewest that fit the riser limit.
The stringer’s bottom riser is cut this much shorter.
Joist + subfloor + ceiling, to size the opening for headroom.
A 2x12 is 11 1/4 in (PS 20, dry).

Risers

14 × 7 11/16 in

7.714 in exactly; 195.9 mm

Treads
13 × 10 in
Total run
10 ft 10 in
Angle
37.6°
Stringer, along the nosings
14 ft 1 in
  • OK: Riser within the 7 3/4 in limit
  • OK: Tread meets the 10 in minimum
  • OK: One flight can climb this high (12 ft 7 in at most)
  • Attention: Treads under 11 in need a 3/4 to 1 1/4 in nosing on solid risers
  • Attention: A handrail is required (four or more risers), 34 to 38 in above the nosings
  • Attention: Open sides more than 30 in above the floor or grade need a guard

One source: International Code Council, publisher of the IRC (2024 R318.7, 2021 R311.7)

Check your local building code before you build: towns and states amend the IRC and adopt different editions, and some set stricter limits.

Why: The IRC allows risers up to 7 3/4 in, so 108 in of rise needs 14 risers of 7.714 in, all the same height.

Laying out the stringer

Framing square: tongue (riser)
7 11/16 in
Framing square: blade (tread)
10 in
Wood left under the notches (throat)
5 1/8 in

Clamp stair gauges on the square at these two marks and step it down the board. Stringer span, size and spacing are structural: they are not worked out here.

Riser marks from the floor

Each mark is worked from the exact riser, so rounding never adds up along the stair.

  1. 1: 7 11/16 in
  2. 2: 15 7/16 in
  3. 3: 23 1/8 in
  4. 4: 30 7/8 in
  5. 5: 38 9/16 in
  6. 6: 46 5/16 in
  7. 7: 54 in
  8. 8: 61 11/16 in
  9. 9: 69 7/16 in
  10. 10: 77 1/8 in
  11. 11: 84 7/8 in
  12. 12: 92 9/16 in
  13. 13: 100 5/16 in
  14. 14: 108 in

Comfort (not code)

  • Attention: 2 risers + tread = 25.43 in (about 24–25 in feels natural). Ask the Builder allows 24–26 in, with 25 the sweet spot.

Traditional rule, stated by Swanson Tool Co. and Ohio University

  • OK: Riser + tread = 17.71 in (17–18 in)

Traditional rule, stated by Fine Homebuilding, JLC and Swanson Tool Co.

Why these numbers?

  1. The situation

    108 in from floor to floor, with 10 in treads.

    From your input: the rise and the tread

  2. What that means

    The IRC limits risers to 7 3/4 in and treads to at least 10 in, with every riser and every tread in a flight within 3/8 in of the others (2024 R318.7.5, 2021 R311.7.5).

    One source: International Code Council, the model building code , sources [1, 2]

  3. So

    Rise ÷ risers = 108 ÷ 14 = 7.7143 in. There is one tread fewer than risers (the upper floor is the last step): 13 × 10 in = 130 in of run.

    Arithmetic: exact by definition

  4. The number

    Angle = atan(7.714 ÷ 10) = 37.6°; along the nosings √(108² + 130²) = 169.01 in.

    Arithmetic: exact by definition

  5. What that means

    Comfort, not code: two risers plus a tread of about 24–25 in (here 25.43 in), and a riser plus a tread of 17–18 in (here 17.71 in).

    Traditional rule: Swanson Tool Co., Ohio University, Ask the Builder (Tim Carter), Fine Homebuilding and JLC (Journal of Light Construction) , sources [3, 4, 5, 6, 7, 3]

  6. The number

    Check your local building code before you build: towns and states amend the IRC and adopt different editions, and some set stricter limits.

    From your input: where you build

How we source these numbers

How a stair is worked out

Start from the rise: measure from the finished floor at the bottom to the finished floor at the top, where the stair will land. Divide it by the tallest riser you will accept, 7 3/4 inches under the IRC, and round up: that is the number of risers. Every riser is then the rise divided by that number, so they all come out the same height. The code lets the tallest and shortest riser in a flight differ by 3/8 inch at most, and a trip is most likely where one step is a different height.

There is one tread fewer than risers, because the upper floor is the last step. The run is the treads times the tread depth, and the stringer's length along the nosings is the hypotenuse of the rise and the run. If the stair has to fit a space, enter the run you have and the treads share it; if they come out under 10 inches, the stair needs more room, since adding risers adds treads too.

To lay out a cut stringer, clamp stair gauges on a framing square at the riser on the tongue and the tread on the blade, and step the square down the board. Cut the bottom riser shorter by the tread thickness so the first step comes out the same height as the rest. The riser marks from the floor are worked from the exact riser, so a 1/16 rounding doesn't add up over fourteen steps.

Stringer span, size and spacing are structural and not worked out here; for deck stairs the American Wood Council's DCA 6 guide covers them, and the deck calculator frames the deck the stair comes down from. The full list of limits is on the stair code requirements page, and the 2x12 actual size page shows why a 2x12 stringer is 11 1/4 inches deep.

Sources

  1. International Code Council, 2024 International Residential Code, Chapter 3: R318.7 Stairways, R320 Handrails, R321 Guards (free read-only text), read 2026-11-07
  2. International Code Council, 2021 International Residential Code, Chapter 3: R311.7 Stairways, R312 Guards (free read-only text), read 2026-11-07
  3. Swanson Tool Co., The Pro Reference (Blue Book), ch. 5: stair stringer layout, read 2026-11-07
  4. Ohio University, ART 2600 stair design handout: 2R + T = 24–25 inches, read 2026-11-07
  5. Ask the Builder (Tim Carter), Best stair layout: two risers plus a tread between 24 and 26 inches, read 2026-11-07
  6. Fine Homebuilding, 2 rules for comfortable stairs; How to cut stair stringers, read 2026-11-07
  7. JLC (Journal of Light Construction), Stair stringers: calculation and layout, read 2026-11-07
  8. NIST, PS 20-25 American Softwood Lumber Standard, Table 3: minimum dressed sizes, read 2026-11-07

How we source these numbers