Stair Math: Risers, Treads & the Code Rules That Keep Them Legal

Q: What is the maximum stair riser height?

Per IRC R311.7.5.1, no riser may exceed 7¾ inches. That ceiling is why a 105″ rise needs 14 risers rather than 13: 105 ÷ 7.75 = 13.5, and you can't build half a riser.

Most stair screw-ups are not framing errors — they're arithmetic errors made before the first cut. Here's the math that matters and the IRC numbers that govern it.

The two numbers that matter

Riser height is the vertical distance from the top of one tread to the top of the next. Tread depth (the "run") is the horizontal distance from the leading edge of one tread to the leading edge of the next — nosing to nosing, not stringer to stringer. The total rise is from the finished floor at the top to the finished floor or landing at the bottom. Measure it after flooring is installed, or your riser heights will be off by the thickness of the flooring.

The IRC limits that actually get checked

Max riser: 7¾″ (R311.7.5.1). Not 8″. The 7¾″ number is why a 105″ rise needs 14 risers, not 13.
Min tread depth: 10″, measured nosing to nosing (R311.7.5.2).
Variation: all risers within 3/8″ of each other, and all tread depths within 3/8″ (R311.7.4.3). One shimmed or mismeasured riser fails inspection even if every step is under 7¾″.
Nosing: ¾″ to 1¼″ beyond the riser below, and nosing projections within 3/8″ of each other (R311.7.5.3).
Clear width: 36″ minimum (R311.7.1).
Headroom: 6′8″ minimum, measured plumb from the nosing line (R311.7.2).
Handrail: required on at least one side for any flight with 4 or more risers, 34″–38″ above the nosing, continuous and graspable (R311.7.8).
Guards: required on open sides where the drop is more than 30″, 36″ minimum height (R312).
Landings: top and bottom, at least as wide as the stairs and 36″ minimum in the direction of travel (R311.7.6).
Open risers: allowed only if the opening won't pass a 4″ sphere (R311.7.4.1).

The comfort rule that isn't in the code

Ergonomic research and stair-design tradition both point at the same band: riser + tread = 17″–18″ (about 17½″ ideal). Notice the code minimums land right in it — 7.75 + 10 = 17.75. A second formula, 2 × riser + tread = 24″–25″, is favored by many framers for exterior stairs. Neither is code, but both are a great sanity check: if your numbers fall outside the band, the stairs will feel wrong even when they pass.

Rule of thumb: divide your total rise by 7.5, round up to a whole number of risers, then divide again to get the actual riser height. Starting from the middle of the band usually lands you in the comfort zone — the code check is then just confirmation.

Worked example — 105″ of rise

Say your deck-to-grade rise measures 105″. Start with the maximum riser:

105 ÷ 7.75 = 13.5 → you can't build half a riser, so round up to 14 risers.

Now divide for the real number: 105 ÷ 14 = 7.5″ per riser — legal (under 7¾″) and right in the comfort band. Treads are always one fewer than risers: 13 treads.

At a 10″ tread that's a 130″ total run (13 × 10 = 130″ = 10′10″) — that's how much floor space the stair actually eats, so check it against the landing before you build. The stringer itself is the hypotenuse: √(105² + 130²) ≈ 167″ ≈ 13′11″, at an angle of about 39°. Buy stringer stock longer than that — the bottom cut and any landing detail eat length.

If the run comes out too long for the space, you can't shrink the treads below 10″. You either push the total rise up (steeper, up to 7¾″ per riser), add a landing and split the flight, or change the geometry of the approach.

Common mistakes

Measuring rise before finish flooring is in. The riser is measured between finished tread surfaces; a ¾″ floor change shifts every riser in the flight.
Using 7¾″ as the riser instead of dividing. The 7¾″ is a ceiling, not a target. You divide the actual total rise by the riser count and use that number.
Ignoring the 3/8″ variation rule. It's the #1 stair inspection failure: every riser and every tread must match within 3/8″, not just stay under the max.
Measuring treads between the stringers. Tread depth is nosing to nosing, and with a nosing the actual board is longer than the run.
Forgetting headroom until it's built. A stair under a low header or sloped ceiling that fails the 6′8″ plumb check is a very expensive fix.
Handrail height measured to the tread. It's 34″–38″ above the nosing line, measured vertically, not along the rake.

Stair math FAQ

What is the maximum stair riser height?

Per IRC R311.7.5.1, no riser may exceed 7¾″. That ceiling is why a 105″ rise needs 14 risers rather than 13: 105 ÷ 7.75 = 13.5, and you can't build half a riser.

How many risers does my stair need?

Divide the total rise by 7.5, round up to a whole number, then divide the total rise by that count to get the actual riser height. Example: 105″ ÷ 14 = 7.5″ per riser, which is legal and sits in the comfortable 17″–18″ riser-plus-tread band.

Why does stair riser variation matter?

IRC R311.7.4.3 requires all risers to be within 3/8″ of each other and all tread depths within 3/8″. It's the most common stair inspection failure — one shimmed or mismeasured riser fails the stair even when every step is under the 7¾″ maximum.

How much headroom do stairs need?

IRC R311.7.2 requires 6′8″ minimum headroom, measured plumb from the nosing line. A stair built under a low header or sloped ceiling that fails this check is a very expensive fix.

Skip the arithmetic

The stair stringer calculator does the whole layout for you — riser count and height, treads, total run, stringer length, angle, and the exact framing-square cuts. Plug in your total rise and tread run and it handles the ceiling-and-divide step that trips everyone up.

And check your local amendments: most jurisdictions adopt the IRC with changes, and some keep the older 8¼″ maximum or require 11″ treads on certain stair types. The numbers above are the baseline — the AHJ's version is the one that counts.