The math behind the stair stringer layout calculator
Eight formulas cover everything the tool shows. Each is worked through for the same stair: a 36 in rise with 48 in of run, 1 in treads and a 2x12 stringer.
Number of risers
N = ⌈ H ÷ 7.5 in ⌉
Worked: 36 ÷ 7.5 = 4.8 → 5 risers
Round up so no riser exceeds your target. The IRC maximum is 7 3/4 in; aiming at 7 1/2 in leaves room for an uneven pad.
Unit rise (riser height)
R = H ÷ N
Worked: 36 ÷ 5 = 7.20 in (7 3/16)
Every riser is identical. This is the number you set on the tongue of the framing square.
Treads and unit run
n = N − 1 · T = total run ÷ n
Worked: 5 − 1 = 4 treads · 48 ÷ 4 = 12.00 in
The deck or upper floor is the top step, so there is one tread fewer than risers. T goes on the body of the square.
Stringer pitch
θ = arctan(R ÷ T)
Worked: arctan(7.20 ÷ 12.00) = 31.0°
The stringer follows the steps, so its angle is one riser over one tread. It is not total rise ÷ total run (36 ÷ 48 gives 36.9°, which is the steeper line from the pad to the deck edge).
Bottom drop
first riser cut = R − tread thickness
Worked: 7.20 − 1.00 = 6.20 in
Every other step gains a tread on top and loses one below; the first only gains one. Cutting it short keeps the bottom step equal to the rest.
Throat
throat = D − (R × T) ÷ √(R² + T²)
Worked: 11.25 − (7.20 × 12.00) ÷ 13.99 = 5.08 in
D is the board depth (11 1/4 in for a 2x12). The second term is how deep each notch reaches into the board, measured square to its edge. AWC DCA 6 asks for at least 5 in left on deck stairs.
Cut length along the board
from the toe of the level cut to the heel of the top plumb cut, measured along the edge
Worked: 59.17 in → buy a 6 ft 2x12
Longer than √(rise² + run²) because the board also has to reach past the bottom level cut and the top plumb cut. Buy the next standard length up so you can trim a split end.
Comfort check (stride)
24 in ≤ 2R + T ≤ 25 in
Worked: 2 × 7.20 + 12.00 = 26.40 in
The Blondel rule: an average stride is about 24–25 in. This example is long-striding because the run is generous; 11 in treads would walk more naturally.
Reverse the math: check a stringer you already have
Old or pre-cut stringers are often reused. To check one, lay a framing square in any notch and read the rise on the tongue and the run on the body. Multiply the rise by the number of risers the stringer will make: that total has to match your measured rise within about 3/8 in, or the top or bottom step will be off. A 7 1/4 in stringer with five risers climbs 36 1/4 in, so on a 36 in deck the bottom drop needs an extra 1/4 in taken off.
Then measure the throat at the deepest notch, square to the bottom edge. If it is under 5 in on a deck stair, add a solid center stringer or replace it.
Rounding to sixteenths
The tool keeps full precision and shows the nearest sixteenth for marking. Rounding every riser to the nearest sixteenth adds at most 1/32 in per step, which is well inside the 3/8 in the IRC allows across a flight. If you set your stair gauges by eye, set them once and do not move them between notches: consistent is better than exact.
Metric
The same formulas work in millimeters. Aim for a riser of 190 mm or less and a tread of at least 254 mm; the stride range becomes 610–635 mm. The tool converts everything when you switch units.
Run your own numbers in the stair stringer layout tool, or see how to mark them on the board.