The maths of a staircase is short. If you want to calculate stairs from a floor-to-floor height, the whole formula fits in four lines and uses nothing beyond division and Pythagoras.
What ruins stairs is not in those four lines. It is a fifth step that lives in carpentry books and almost never in stair calculators — and skipping it leaves your first step taller than every other one, by more than three times the tolerance the code allows.
Here is the whole thing, followed by the part that gets left out.
The Formula, in Four Lines
Everything starts from one measurement: the total rise, measured from the finished floor at the bottom to the finished floor at the top. Not the subfloor — the surface you will actually walk on, or your whole stair will be out by the thickness of your flooring.
1. risers = round(total rise ÷ your target riser height)
2. riser height = total rise ÷ risers
3. total run = (risers − 1) × tread depth
4. stringer = √(total rise² + total run²)
angle = arctan(total rise ÷ total run)Line 1 is the only place you make a choice. You pick a riser height that feels right — around 7 inches or 175 mm is the usual starting point — and round to a whole number, because you cannot build two-thirds of a step.
Line 2 is the one that matters most. You divide back. Whatever riser height you were aiming for, the real one is the total rise divided by the whole number of risers, and that is what makes every step identical. Stairs are dangerous when steps differ, and the human foot notices a few millimetres.
A worked staircase calculation example, in inches:
total rise 108″, aiming for 7″ risers
108 ÷ 7 = 15.43 → 15 risers
108 ÷ 15 = 7.2″ per riser
(15 − 1) × 10″ = 140″ of run
√(108² + 140²) = 176.8″ of stringer
arctan(108 ÷ 140) = 37.6°That is the stair calculation formula in inches from end to end. In millimetres it is identical — only the numbers you feed it change.
Why There Is Always One More Riser Than Tread
Line 3 subtracts one, and that catches people out constantly. If you have 15 risers you have 14 treads, always.
The reason is that the last riser does not land on a tread. It lands on the floor above. Climb the stair in your head: you step up 15 times, but you only ever stand on 14 boards, because the fifteenth step puts you on the landing.
Get this wrong and your run is out by a full tread — 250 mm or 10 inches of floor space you either did not budget for or wasted. It is the single most common arithmetic slip in stair layout.
The Fifth Line Nobody Writes Down
Now the part the four lines do not cover.
Your stringer gets cut with identical notches, each one riser high. It sits on the lower floor. Then tread boards go on top of each notch — and that is where it goes wrong. The stringer sits on the floor, so the first notch already starts at floor level, and putting a tread on it lifts that first step by the full thickness of the board.
Every other step is fine, because each one has a tread below it as well as above it. Only the bottom step has floor underneath instead of a tread.
riser 180 mm, tread board 32 mm
every step = 180 mm
the first step = 180 + 32 = 212 mmThirty-two millimetres taller than the rest. That is not a rounding error — that is the sort of difference that catches a toe.
And it is a code violation. The IRC allows the tallest and shortest riser in a flight to differ by no more than 3/8 of an inch, which is 9.5 mm. Here is how far common tread thicknesses push you past that:
| Tread board | First step becomes | Over the 9.5 mm limit by |
|---|---|---|
| 19 mm (¾″) | 199 mm | 9.5 mm over — twice the allowed variation |
| 25 mm (1″) | 205 mm | 15.5 mm over |
| 32 mm (1¼″) | 212 mm | 22.5 mm over — over three times the allowed variation |
| 38 mm (1½″) | 218 mm | 28.5 mm over — four times the allowed variation |
The fix is one line, and carpenters call it dropping the stringer:
5. bottom cut = riser height − tread thicknessYou cut that much off the bottom edge of the stringer before you fit it. The first notch then sits one tread thickness lower, the tread brings it back up, and every step in the flight measures the same.
There is a refinement worth knowing. If finished flooring still has to go down at the bottom of the stair, that flooring will raise the walking surface later. So the drop is really tread thickness − floor finish thickness. And if the floor finish is thicker than your tread — thick tile against a thin tread, say — the number goes negative, which means the stringer needs packing up rather than cutting down.
The Comfort Rule, and Its Two Disguises
Riser height and tread depth are not independent. A tall riser wants a shallow tread and a shallow riser wants a deep one, because your stride length does not change just because the stair did.
That relationship is written down in two ways, and most people meet only one of them:
| Where | Written as | Range | Status |
|---|---|---|---|
| UK, Australia, NZ | 2R + G | 550–700 mm | In the code |
| United States | 2R + T | 24–25 inches | Rule of thumb only |
They are the same idea. Convert the American band and something interesting falls out: 24 inches is 610 mm and 25 inches is 635 mm, so the 2R + T band sits entirely inside the 550–700 mm one.
Which means a stair that satisfies the American rule of thumb automatically satisfies the British and Australian codes. The reverse is not true — plenty of perfectly legal UK stairs sit outside the tighter American comfort band.
Rearranged, the rule tells you what tread to pair with a riser you have already settled on:
| Riser height | Tread depth that keeps 2R + G in range |
|---|---|
| 150 mm | 250 – 400 mm |
| 165 mm | 220 – 370 mm |
| 175 mm | 200 – 350 mm |
| 180 mm | 190 – 340 mm |
| 190 mm | 170 – 320 mm |
| 200 mm | 150 – 300 mm |
Those bands satisfy the comfort rule and nothing else. Your code's own minimum tread still applies on top — 220 mm in the UK, 240 mm in Australia, 254 mm under the IRC — so the shallow end of each row is often ruled out before you get there. A 200 mm riser is itself over the Australian maximum of 190 mm.
One Rise, Four Codes, Four Different Staircases
Here is where it stops being arithmetic and starts being geography.
I took one ordinary floor-to-floor height — 2,700 mm, about 8 feet 10 inches — and worked out the most compact legal staircase under each of the four codes the calculator supports — the one with the fewest risers, and so the shortest run — each using the tread depth normal in that country. Same climb, same house, four different answers:
| Code | Fewest legal risers | Riser | Tread | Run | Angle |
|---|---|---|---|---|---|
| US · IRC (residential) | 14 | 192.9 mm | 254 mm (10″) | 3,302 mm | 39.3° |
| US · IBC (commercial) | 16 | 168.8 mm | 279.4 mm (11″) | 4,191 mm | 32.8° |
| UK · Part K | 15 | 180.0 mm | 225 mm | 3,150 mm | 40.6° |
| AU / NZ · NCC | 15 | 180.0 mm | 250 mm | 3,500 mm | 37.6° |
The run swings from 3,150 mm to 4,191 mm. That is a full metre of floor space — over three feet — separating the same climb built in London from the same climb built to American commercial code. If you are laying out a plan, that is not a detail. That is a room.
Two things in that table are worth pulling out.
The UK figure is not set by the riser limit. Part K allows a rise up to 220 mm, which would let 2,700 mm be climbed in 13 risers. But Part K also caps the pitch at 42°, and 13 risers with a 225 mm going comes out at 45°. Fourteen is still 42.7°. It takes 15 risers to get under the line. The pitch rule, not the riser rule, is what decides.
A 250 mm tread is illegal in an American home. Two hundred and fifty millimetres is the standard going across Australia and perfectly normal in Britain. The IRC minimum tread is 10 inches — 254 mm. A 250 mm tread misses it by four millimetres. Under the commercial IBC, where the minimum is 11 inches or 279.4 mm, it misses by twenty-nine.
What Each Extra Riser Costs You in Floor
People pick a riser count for comfort and then discover the stair does not fit. The relationship is simple and worth internalising: every extra riser adds exactly one tread of run.
Same 2,700 mm rise, IRC, 10-inch treads:
| Risers | Riser height | Run | Angle |
|---|---|---|---|
| 14 | 192.9 mm (7.59″) | 3,302 mm (10.83 ft) | 39.3° |
| 15 | 180.0 mm (7.09″) | 3,556 mm (11.67 ft) | 37.2° |
| 16 | 168.8 mm (6.64″) | 3,810 mm (12.50 ft) | 35.3° |
| 17 | 158.8 mm (6.25″) | 4,064 mm (13.33 ft) | 33.6° |
| 18 | 150.0 mm (5.91″) | 4,318 mm (14.17 ft) | 32.0° |
| 19 | 142.1 mm (5.59″) | 4,572 mm (15.00 ft) | 30.6° |
Going from 14 risers to 19 makes the stair markedly gentler — 39.3° down to 30.6° — and costs 1,270 mm of extra floor. Four feet two inches. That is the trade the whole layout turns on, and it is why the riser count is worth stepping up and down through rather than accepting the first number a calculator gives you.
The Hole in the Floor Above
One measurement gets forgotten until the framing is up: the opening in the upper floor has to be long enough that nobody walking under it hits their head.
Codes want roughly two metres of clearance, measured vertically from the tread nosings — 2,000 mm in the UK and Australia, 6 feet 8 inches in the United States, which is 2,032 mm. To find the opening you need, work out how far the stair must descend to clear the underside of the floor above, then convert that into treads:
floor opening = tread depth × (headroom + floor thickness) ÷ riser heightFor that same 2,700 mm stair at 180 mm risers and 250 mm treads, with a 300 mm floor structure and 2 m of headroom, the opening comes out at 3,194 mm — close to 10 feet 6 inches. The American equivalent, a 108-inch rise under IRC with a 12-inch floor, needs about 3,029 mm, just under 10 feet.
Either way it is longer than most people picture. A stairwell is not a hatch.
Working It Out on CodBolt
- Open the Stair Calculator and pick your building code — IRC, IBC, UK Part K, the Australian and New Zealand NCC, or no code at all if you only want the geometry.
- Enter the total rise. Inches, feet and inches, millimetres and centimetres all work, and switching between them converts what you have typed rather than clearing it.
- The riser count is chosen for you and capped so it never breaks your code's maximum. Step it up or down with the buttons to see the run and angle move, exactly as in the table above.
- Open Stringer & headroom details if you are cutting timber. That is where the tread thickness lives, and where the bottom cut, the throat left behind the notch and the floor opening are worked out.
- The code check lists every limit separately, and marks the comfort rule as guidance rather than law where it is not actually in the code.
It is geometry, so it all runs in the page — nothing is uploaded, and it keeps working with the network off.