Stair math starts and ends with one measurement — the total rise — divided into equal steps that pass code and feel natural underfoot. Get the riser height right and the stringer, run, and materials all fall out of it. This guide runs from riser counts to stringer layout, codes, and materials — distilling the 100 questions builders and DIYers ask most into one readable pass.
Four terms describe any staircase. Total rise is the vertical floor-to-floor height; total run is the horizontal distance it covers. Unit rise is one step's height; unit run is one step's depth (riser face to riser face, not counting nosing). The whole calculation flows from dividing the total rise into equal risers.
There's always one fewer tread than riser because the top "tread" is the upper floor itself. Blondel's rule is the ergonomic sanity check — two risers plus a tread should land in the natural-stride window of 24–25 inches.
108 ÷ 7 = 15.4 → 15 risers, each 108 ÷ 15 = 7.2″. That's 14 treads; at a 10″ run the total run is 140″ (11.7 ft). Blondel: 2 × 7.2 + 10 = 24.4″ ✓.
Codes exist because uneven or steep stairs are a leading fall hazard. Residential (IRC) and commercial (IBC) limits differ, so check which applies:
| Parameter | IRC (residential) | IBC (commercial) |
|---|---|---|
| Max riser height | 7¾″ | 7″ |
| Min tread depth | 10″ (11″ no nosing) | 11″ |
| Min width | 36″ | by occupancy load |
| Min headroom | 6 ft 8″ (80″) | 6 ft 8″ |
| Handrail height | 34–38″ | 34–38″ |
Two more rules a calculator tracks: the 3/8″ variance rule — the tallest and shortest riser (or deepest and shallowest tread) in a flight can't differ by more than 3/8″ — and the landing rule: no straight flight may exceed 12 ft of vertical rise without a landing, and a landing must be as wide as the stair and at least 36″ deep in the direction of travel.
The stringer is the diagonal 2×12 that carries the steps. Its raw length is the hypotenuse of the rise/run triangle:
Cut stringers must keep a minimum 5″ throat (solid wood below the notches) so they don't split. Mark the steps fast with stair gauges clamped to a framing square at your unit rise and run. And do the single most-missed adjustment — "dropping the stringer."
Before fastening, trim the bottom of the stringer by exactly the tread thickness (e.g. 1½″). Skip it and the first step ends up a tread-thickness too tall and the top step too short — a trip hazard and a code fail on the 3/8″ variance.
An L-shaped stair turns 90° at a landing; a U-shaped (switchback) makes a 180° turn at a wide landing. The method is the same: treat it as two independent flights — flight 1 from the lower floor to the landing height, flight 2 from the landing to the upper floor — and calculate each. Ideally place the landing to split the rise evenly so both flights share the same step count and unit rise. Watch the landing's finish thickness: if it differs from the steps, adjust the stringer feet where they land.
Turning stairs without a landing use special geometry. Winder steps are triangular — wider on the outside, narrow at the inside corner. Code measures their depth along a walkline 12″ out from the inside rail, where they must hit the normal minimum (e.g. 10″), and they must keep at least 6″ of depth at the narrowest point. Spiral stairs rotate 22.5–30° per step, so a full 360° loop takes 12–16 steps, and the footprint is a cylinder (π × radius² × height).
The well-hole is the opening cut in the upper floor so people clear the ceiling as they climb. To size it, trace the sloped headroom line down from the ceiling and find where the gap from the nosing to the ceiling drops to exactly 6 ft 8″ (80″) — that point sets the minimum length of the floor cutout. A steeper stair drops faster, so it needs a shorter well-hole; a shallow stair needs a longer one. Thick double header joists eat into clearance, so widen the opening if needed, and on a stair under a sloped roofline, plot the path against the ceiling to keep 80″ everywhere along the walkline.
Open sides need a guardrail. On the stair itself it's a minimum 34″ (measured from the nosings); on flat upper landings it's 36″ residential, 42″ commercial. The two "sphere" rules govern infill spacing:
How many stringers depends on width — support every ~16 inches so treads don't sag:
| Width | Stringers |
|---|---|
| 36–48″ (residential) | 3 (left, center, right) |
| 72″ (commercial) | 5 |
Use straight, knot-free #2-or-better structural lumber (Southern Yellow Pine or Douglas Fir) — a crowned 2×12 throws off every step. Exterior stairs need ground-contact pressure-treated wood. For treads, buy 3-ft pre-cut planks or cross-cut 12-ft blanks into three 36″ pieces. Plan about 1 tube of construction adhesive per 4 steps (glue plus screws prevents squeaks), and fasten treads with 2½″ deck screws or 3″ ring-shank nails. Pre-cut stringer modules (7″ rise / 11″ run) suit quick 3–4 step deck runs, and metal stringers are estimated by rise and step count.
For a poured concrete stair, the bulk is a triangular wedge plus the step "teeth":
Commercial steel pan stairs use pre-formed steel trays welded into channel stringers, filled on site with ~1.5–2″ of concrete per step — estimate the channel linear footage, pan count, and fill volume. Commercial egress also sizes width to occupancy: about 0.3″ of width per person. Steeper specialty options (ship's ladders at 50–70°, alternating-tread stairs) are code-limited to tight access.
Total material cost stacks stringers, treads, risers, handrails, and balusters against local prices, plus delivery and tax. A few finishing items to include:
Re-run the step math, verify the layout forms are square with a laser, and cross-check your material order before cutting stringers or pouring concrete. Then measure each finished riser to confirm it's within the 3/8″ code limit before the stair goes into service.