# Stair calculator

> Number of steps, riser height, tread depth, stringer length and stair angle from total rise and run, checked against IRC riser and tread limits.

Interactive version: https://www.calcopenly.com/construction/stair-calculator
Subject: Construction and home calculators

The number of risers is the floor-to-floor rise divided by the tallest riser you will accept, rounded up, so every riser is the same height and none exceeds your limit. The run is divided by one fewer than the risers to give the tread depth, because the upper floor forms the top step. The pitch is arctan(riser ÷ tread), the slope of the line through the nosings, and the stringer length is √(rise² + run²).

The default stair, 108 in (9 ft) floor to floor over 140 in of run, gives 15 risers of 7.2 in and 14 treads of 10 in at a pitch of 35.8°. That meets the US residential limits of a 7¾ in maximum riser and 10 in minimum tread (2021 IRC R311.7.5; 2024 IRC R318.7.5).

Measure the rise between finished floor surfaces. Tread depth here is the going, nosing to nosing. The check covers risers and treads only; stair width, headroom, handrails and local amendments need checking separately.

## Inputs

- **Total rise (floor to floor)**
- **Total run available**
- **Tallest riser you'll accept**: The stair uses the fewest risers that keep each one at or under this height.

## Results

- Risers — main result
- Riser height (in)
- Riser height (mm)
- Treads
- Tread depth (in)
- Tread depth (mm)
- Stringer length (in)
- Stringer length (ft)
- Stair angle (pitch) (°)
- IRC riser and tread check

## Formula

$$
n = \left\lceil \frac{\text{rise}}{r_{\max}} \right\rceil,\quad r = \frac{\text{rise}}{n},\quad t = \frac{\text{run}}{n - 1},\quad \theta = \arctan\frac{r}{t},\quad s = \sqrt{\text{rise}^2 + \text{run}^2}
$$

## Worked examples

### 9 ft floor to floor, 140 in of run

- Total rise (floor to floor): 108 in
- Total run available: 140 in
- **Risers: 15**
- **Riser height: 7.2 in**
- **Treads: 14**
- **Tread depth: 10 in**
- **Stringer length: 176.82 in**
- **Stair angle (pitch): 35.8 °**
- **IRC riser and tread check: Meets IRC riser and tread limits**
- Checked against: Python decimal: ⌈108/7.5⌉ = 15; 108/15 = 7.2 in; 140/14 = 10 in; √(108² + 140²) = 176.816 in; pitch atan(7.2/10) = 35.754°, pitch being the slope of the line through the nosings (Approved Document K, 2013, Appendix A)

### Edge: 3-4-5 stair lands on an exact riser count

- Total rise (floor to floor): 105 in
- Total run available: 140 in
- **Risers: 14**
- **Riser height: 7.5 in**
- **Tread depth: 10.769 in**
- **Stringer length: 175 in**
- **Stringer length: 14.58 ft**
- Checked against: Hand calculation: 105/7.5 = 14 exactly; 140/13 = 10.769 in; 105:140:175 is a 3-4-5 triangle

### Short run fails the tread minimum

- Total rise (floor to floor): 100 in
- Total run available: 100 in
- Tallest riser you'll accept: 7.75 in
- **Risers: 13**
- **Riser height: 7.692 in**
- **Tread depth: 8.333 in**
- **Stair angle (pitch): 42.7 °**
- **IRC riser and tread check: Tread under 10 in**
- Checked against: Python decimal: ⌈100/7.75⌉ = 13; 100/12 = 8.33 in < 10 in (2021 IRC R311.7.5.2); pitch atan(7.6923/8.3333) = 42.709°

### Metric: 2800 mm rise, 3600 mm run, 180 mm target

- Total rise (floor to floor): 2800 mm
- Total run available: 3600 mm
- Tallest riser you'll accept: 180 mm
- **Risers: 16**
- **Riser height: 175.0 mm**
- **Tread depth: 240.0 mm**
- **IRC riser and tread check: Tread under 10 in**
- Checked against: Hand calculation: ⌈2800/180⌉ = 16; 2800/16 = 175 mm; 3600/15 = 240 mm < 254 mm

### Steep ladder-like stair fails both limits

- Total rise (floor to floor): 120 in
- Total run available: 90 in
- Tallest riser you'll accept: 8 in
- **Risers: 15**
- **Riser height: 8 in**
- **Tread depth: 6.429 in**
- **IRC riser and tread check: Riser over 7¾ in and tread under 10 in**
- Checked against: Hand calculation: 120/8 = 15 risers of 8 in > 7¾ in; 90/14 = 6.43 in < 10 in

## Questions

### What is the maximum stair riser height allowed by code?

In US homes it is 7¾ in, with treads at least 10 in deep and no more than 3/8 in difference between the tallest and shortest riser in a flight (2021 IRC R311.7.5.1–R311.7.5.2; 2024 IRC R318.7.5). Buildings under the 2021 IBC (§1011.5.2) need risers of 4 to 7 in and treads of at least 11 in, except R-3 buildings and stairs inside R-2 dwelling units, which may use 7¾ in and 10 in.

### How many steps do I need for a 9 ft rise?

At least 14 risers. Divide the rise by the maximum riser and round up: 108 in ÷ 7.75 in = 13.9, so 14 risers of 7.71 in; a 7.5 in target gives 15 risers of 7.2 in. There is one tread fewer than risers (13 or 14), because the upper floor acts as the last step. Measure from finished floor to finished floor, including flooring.

### How do you calculate stair stringer length?

Apply Pythagoras to the total rise and run: √(rise² + run²). For 108 in of rise over 140 in of run that is √(108² + 140²) = 176.8 in, or 14.73 ft, along the pitch line. The plumb cut at the top and the level cut at the bottom need extra board, so buy the next stock length up, 16 ft here.

### What is the steepest stair angle allowed?

For US homes, the IRC's 7¾ in riser over a 10 in tread gives a maximum of arctan(7.75/10) ≈ 37.8°. IBC stairs, at 7 in over 11 in (2021 IBC §1011.5.2), top out near 32.5°. England's Approved Document K, Table 1.1, allows 42° for a private stair, with rises of 150–220 mm, goings of 220–300 mm and twice the rise plus the going between 550 and 700 mm.

### What other rules apply to residential stairs?

The IRC stairway section (2021 R311.7; 2024 R318.7) requires a clear width of at least 36 in and headroom of 6 ft 8 in. Handrails go 34 to 38 in above the tread nosings (2021 IRC R311.7.8; 2024 IRC R320). Riser heights and tread depths may differ by at most 3/8 in within one flight, so cut every step from the same layout.

### How accurate is the stair calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 5 worked examples whose answers come from independent sources; for example, “9 ft floor to floor, 140 in of run” is checked against Python decimal: ⌈108/7.5⌉ = 15; 108/15 = 7.2 in; 140/14 = 10 in; √(108² + 140²) = 176.816 in; pitch atan(7.2/10) = 35.754°, pitch being the slope of the line through the nosings (Approved Document K, 2013, Appendix A).

### Where does the method come from?

City of Boise code handout #437 — IRC R311.7.5 stair treads and risers (riser 7¾ in max, tread 10 in min); International Code Council — 2021 IRC R311.7 stairways (renumbered R318.7 in the 2024 IRC); HM Government — Approved Document K (England), Table 1.1: rise, going and pitch of private stairs.

## Sources

- [City of Boise code handout #437 — IRC R311.7.5 stair treads and risers (riser 7¾ in max, tread 10 in min)](https://www.cityofboise.org/media/14405/437-treads-risers_march-2022.pdf)
- International Code Council — 2021 IRC R311.7 stairways (renumbered R318.7 in the 2024 IRC)
- [HM Government — Approved Document K (England), Table 1.1: rise, going and pitch of private stairs](https://www.gov.uk/government/publications/protection-from-falling-collision-and-impact-approved-document-k)