A cut stringer is a 2x12 with a row of triangles taken out of it, and every one of those triangles has to be the same. This calculator does the arithmetic once (riser count, unit rise, run, throat, board length and where each step lands on the board) so you can pencil the whole stringer from a single hook of the tape and trace the rest from the first one. A 36" deck works out to 5 risers at 7-3/16" on 4 treads, a 40" run and an 8 ft 2x12.
Total rise is the measurement everything hangs on
Measure from the finished deck surface down to the finished surface the bottom step will land on, not straight down from the rim. Grade usually falls away from a house, so take the reading roughly one total run out from the deck, where the landing will actually be. If the ground slopes there, set the landing first and measure to it.
Divide the rise by 7-3/4" and round up: that is the fewest risers the code allows. Divide the rise by that count for the unit rise. Each extra riser makes the stair gentler and adds one tread of run.
risers = ⌈ total rise ÷ 7.75 ⌉ unit rise = total rise ÷ risers treads = risers − 1 run = treads × unit run
Why the bottom riser is cut short
Every step on the stringer gets a tread on top of it except the landing. If you lay out 5 equal 7-3/16" risers and nail treads on, the first step finishes one tread thickness taller than the rest, and the top step one tread shorter. Cut the tread thickness off the bottom of the stringer instead: 6-3/16" for 1" 5/4 or composite treads, 5-11/16" for 1-1/2" 2x6 treads.
Skipping the deduction is the classic failed inspection: a 1-1/2" difference is four times the 3/8" the IRC allows between risers.
bottom riser on the stringer = unit rise − tread thickness
The throat holds the stair up
The throat is the uncut wood left between the inside corner of each notch and the bottom edge of the board. AWC DCA 6 requires at least 5" on a cut stringer. A 2x12 at 7-3/16" × 10" keeps 5-7/16", but deep treads at full rise eat into it: 7-3/4" on 11" treads leaves only 4-15/16".
This is also why DCA 6 asks for 2x12 stringers. The same 36" layout on a 2x10 leaves 3-7/16" of throat.
throat = board depth − R × T ÷ √(R² + T²) 2x12 depth = 11-1/4"
Hook the tape once
Stepping a framing square down a board adds every small error to the next. Each step is a diagonal of √(R² + T²), so pencil the steps as running marks from one hook instead: mark 0 where the bottom riser line meets the top edge, then every diagonal from that same zero. Each mark is rounded to 1/16" once, so nothing compounds.
On a 108" basement stair the step-off is 12-5/8". Stepping it 13 times with a rounded diagonal drifts by roughly an eighth; running marks from one hook stay within a thirty-second.
Stringer count, spacing and span
DCA 6 wants at least three cut stringers on a 36" stair, no more than 18" apart for 2x or 5/4 treads. Composite stair spans are set by each board maker, usually 9" to 12", so read the install guide before you lay out. A 36" stair at 16" on centre takes four stringers.
A cut stringer may span 6'-0" horizontally between supports under DCA 6; solid (uncut) stringers go to 13'-3". Past 6 ft, set a 4x4 post on a 12" round or 10" square footing under the stringers, or add a landing. DCA 6 requires an intermediate landing when the total rise passes 12'-0".
Top and bottom connections
At the top, either hang each stringer from the rim with a sloped stair hanger rated at least 625 lb (DCA 6 Fig. 31), such as a Simpson LSCZ, or seat the stringers on a 2x header bolted below the rim. The calculator gives the header height for your layout.
At the bottom, DCA 6 has the stringers bear on a 2x4 block fixed to posts on footings at least 12" below grade or the frost line. A slab that is not founded that deep can be the walking surface, but not the bearing.
The three numbers the inspector checks first
The IRC caps a riser at 7-3/4", wants every tread at least 10" deep measured nosing to nosing, and allows no more than 3/8" between the tallest and shortest riser (or the deepest and shallowest tread) in a flight. That last rule is the one that catches hand-built stairs, almost always at the bottom step.
Section numbers here are the 2021 IRC (R311.7). The 2024 edition keeps the same limits and moves the stair sections to R318. Your local adoption and amendments, and your inspector, have the last word.