Key takeaways
- Convert feet-inches to total inches, calculate in decimals, convert back at the end. Never mix units mid-calculation.
- Riser count comes from dividing total rise by ~7" and rounding; the actual riser is then total rise ÷ that whole number.
- Three proportion rules together: rise + run = 17–18", 2R + T = 24–25", R × T = 72–75.
- Treads always number one fewer than risers, and the bottom of the stringer is dropped by one tread thickness.
A carpenter who miscuts a stud loses a stud. A carpenter who miscuts a stringer loses a 16-foot board and half a day — and if the error is small enough to go unnoticed, it produces a stair with one riser a quarter inch out, which people trip on for the next forty years.
Two things make stairs hard. The arithmetic is in a mixed base — feet are twelve, fractions are sixteenths — and the total rise is fixed by the building, so every dimension has to bend around a number you cannot change.
Feet-inch arithmetic without errors
The reliable method is always the same three steps.
Total inches = (feet × 12) + inches + (numerator ÷ denominator)
9′ 3″ → (9 × 12) + 3 = 111″ · 6′ 4½″ → (6 × 12) + 4 + 0.5 = 76.5″
Convert down, calculate, convert up. Turn every dimension into total decimal inches. Do all the multiplication and division there. Only convert back to feet and inches at the very end, once — because every conversion is a chance to make an error.
Going back the other way: divide total inches by 12 for the feet, and the remainder is the inches.
194.22″ ÷ 12 = 16 remainder 2.22 → 16′ 2¼″
To convert a decimal to sixteenths, multiply by 16: 0.22 × 16 = 3.5 → round to 4/16 = ¼″
Keep the common decimals in your head and most of this becomes instant: ⅛ = 0.125, ¼ = 0.25, ⅜ = 0.375, ½ = 0.5, ⅝ = 0.625, ¾ = 0.75, ⅞ = 0.875. Sixteenths are the odd numbers between them: 1/16 = 0.0625, 3/16 = 0.1875, and so on.
The terms, before the numbers
- Total rise — finished floor to finished floor. The one dimension you cannot alter.
- Total run — the horizontal distance the stair occupies.
- Riser — the vertical height of one step.
- Tread — the horizontal depth of one step, measured nose to nose (the run), not including the nosing overhang.
- Stringer — the sloped board carrying the treads and risers.
- Nosing — the tread's overhang past the riser below, typically ¾″ to 1¼″.
Two facts do all the work. Treads are always one fewer than risers, because the top riser lands on the finished floor above and needs no tread. And every riser must be identical — most codes allow no more than ⅜″ variation between the tallest and shortest riser in a flight, because the human gait calibrates on the first two steps and does not re-check.
The three proportion rules
Comfort is not a matter of taste. Three empirical relationships, all still in use, describe a stair that walks well:
Rise + Run = 17″ to 18″
2 × Rise + Run = 24″ to 25″
Rise × Run = 72 to 75
A stair satisfying all three will feel right. One that fails two of them will not, regardless of whether it passes code.
Code sets the outer limits rather than the comfortable range. The IRC caps the riser at 7¾″ and requires a minimum 10″ tread, with 6′-8″ headroom. In India, the National Building Code recommends a riser no greater than about 190 mm and a tread of at least 250 mm for residential stairs. Both are boundaries, not targets — a stair built at exactly the code limit is legal and unpleasant.
Worked example: a 9′ 3″ floor-to-floor stair
Finished floor to finished floor is 9′ 3″. Tread stock is 1″ thick. Target riser about 7″.
- Convert the total rise. (9 × 12) + 3 = 111″
- Estimate the riser count. 111 ÷ 7 = 15.86 → round to 16 risers
- Find the actual riser. 111 ÷ 16 = 6.9375″ = 6 15/16″
(0.9375 × 16 = 15, so exactly fifteen sixteenths — a clean layout number.) - Count the treads. 16 − 1 = 15 treads
- Choose the run from the rules. Using 2R + T = 24.5: T = 24.5 − (2 × 6.9375) = 24.5 − 13.875 = 10.625″ = 10 5/8″
- Check all three rules.
Rise + Run = 6.9375 + 10.625 = 17.56″ — within 17–18
2R + T = 24.5″ — within 24–25
R × T = 6.9375 × 10.625 = 73.7 — within 72–75 - Total run. 15 × 10.625 = 159.375″ → 159.375 ÷ 12 = 13 remainder 3.375 = 13′ 3⅜″
- Stringer length, by Pythagoras on total rise and total run:
√(111² + 159.375²) = √(12,321 + 25,400) = √37,721 = 194.22″ = 16′ 2¼″ - Order the stock. Add for the top and bottom cuts and for error — order 17′ boards.
- Drop the stringer. Cut the bottom riser at 6.9375 − 1.0 = 5 15/16″, so that once the 1″ treads are on, the first riser matches the other fifteen.
Step 10 is the one that catches people. The stringer is laid out with sixteen equal riser cuts, and then the bottom one is shortened by exactly one tread thickness. Skip it and the bottom step stands an inch proud of every other — well outside the ⅜″ tolerance, and immediately noticeable underfoot.
| Dimension | Decimal inches | Feet-inches | Metric |
|---|---|---|---|
| Total rise | 111.000 | 9′ 3″ | 2819 mm |
| Riser (× 16) | 6.9375 | 6 15/16″ | 176 mm |
| Tread run (× 15) | 10.625 | 10 5/8″ | 270 mm |
| Total run | 159.375 | 13′ 3⅜″ | 4048 mm |
| Stringer length | 194.22 | 16′ 2¼″ | 4933 mm |
| Bottom riser cut | 5.9375 | 5 15/16″ | 151 mm |
The same stair in three notations. 176 mm riser and 270 mm going also satisfy NBC residential guidance.
Where stairs go wrong
- Measuring subfloor to subfloor. The total rise is finished floor to finished floor. If the upper floor gets 20 mm of tile and the lower gets carpet, ignoring both makes every riser wrong.
- Rounding the riser. 111 ÷ 16 = 6.9375, not "about 7". Sixteen risers at 7″ is 112″ — one inch of error that has to go somewhere, and it will all land in the top or bottom step.
- Counting treads as risers. Fifteen treads, sixteen risers. Cutting sixteen tread notches gives a stair that overshoots the run by 10 5/8″.
- Forgetting the drop. As above — the single most common stringer error.
- Losing headroom. 6′-8″ measured vertically from the tread nosing to the ceiling or opening above. Check it at the tightest point before cutting, not after.
- Measuring the tread including nosing. The run is nose to nose. The nosing overhang is comfort, not run, and counting it makes the stair steeper than the numbers suggest.
Frequently asked questions
How do I add feet and inches reliably?
Convert everything to total inches, work in decimals, convert back once at the end. Feet are base 12 and fractions are base 16, so mixing units mid-calculation is where errors come from.
How many risers for my floor height?
Divide total finished floor-to-floor rise by about 7″ and round to the nearest whole number, then divide the total rise by that number for the actual riser height. The total rise is fixed, so the riser must absorb the remainder.
What makes a stair comfortable?
Satisfying all three rules at once: rise + run between 17 and 18 inches, 2R + T between 24 and 25, and R × T between 72 and 75. Code limits — 7¾″ maximum riser, 10″ minimum tread under the IRC — are boundaries, not targets.
Why is the bottom of the stringer cut shorter?
It is called dropping the stringer. The bottom riser is reduced by one tread thickness because the first step gains that thickness when the treads go on. Without it, the bottom riser ends up a full tread thickness taller than the rest.