What a stringer is, and the two ways to build one
A stringer is the sloped board that carries the treads. There are two families, and they behave very differently.
A cut stringer (also called a sawtooth or notched stringer) has the steps sawn out of the top edge, so each tread sits directly on a horizontal cut. It is the standard for exterior deck stairs and for open-sided interior flights. Its weakness is obvious once you draw it: cutting the teeth out removes material, and what is left — the throat — is the only thing carrying load down the slope. Every notch is also a stress raiser at the inside corner.
A solid or housed stringer keeps the full board and supports the treads on cleats, on steel brackets, or in routed housings wedged from behind. It is much stronger for the same depth of board and is what you find on traditional housed-and-wedged interior stairs. It also takes more work.
This calculator lays out a cut stringer, which is what most people are building, and tells you when the geometry has pushed the throat below the point at which a cut stringer is still appropriate. The numbers you need are the diagonal length of the board, the position of every notch, the shortened bottom cut, and the throat.
The geometry: why the throat depends only on riser, run and board width
Lay the board on the bench and mark the layout with a framing square set to the riser on one leg and the unit run on the other. Each step-off advances the square one riser vertically and one unit run horizontally, so the inside corners of the notches — the roots — all fall on a single straight line whose slope is R/G. That line is the spine of the stringer.
The cut length is the diagonal from the bottom corner of the first riser to the top corner where the last riser meets the upper floor. Horizontally that spans the total run, vertically the total rise, so L = √(rise² + run²) by Pythagoras. On a 112 in rise with a 157.5 in run, that is √(12,544 + 24,806.25) = 193.26 in. You buy longer stock than that, because the bottom seat cut and the top plumb cut both extend past those corners.
The throat falls out of the same picture. Every tooth peak sits the same perpendicular distance above the root line, and that distance is the area of the step rectangle divided by its diagonal: R·G ÷ √(R² + G²). Subtract it from the board width and you have the throat. Notice what is not in that expression: the number of steps and the total rise. A three-step stair and a sixteen-step stair with the same riser and run leave exactly the same throat. Notice also that the throat shrinks as either the riser or the run grows — a 7 in riser on a 10 in run leaves 5.52 in of a 2x12, while a 7.75 in riser on an 11 in run leaves only 4.91 in.
The bottom cut needs a separate adjustment that has nothing to do with trigonometry. Every tread except the first sits on top of a notch, and the notch below it is one riser lower, so the surface-to-surface rise is uniform. The bottom of the stringer, though, sits on the pad with no tread beneath it. If you cut a full riser there, adding a 1.5 in tread on top makes the first step 1.5 in taller than the rest. So you shorten the bottom cut by exactly the tread thickness: bottom cut = R − tread thickness. This is the single most common stringer error, and it fails inspection every time because it breaks the 3/8 in riser-uniformity limit in IRC R311.7.5.1.
Worked example: a 45 in deck stair on 2x12 stringers
You have a deck 45 in above the pad. The rise and run calculator gives six risers, so:
- Riser height. 45 ÷ 6 = 7.500 in.
- Treads. 6 − 1 = 5 notches. The deck surface is the sixth step up.
- Total run. 5 × 10.5 = 52.5 in.
- Diagonal of one step. √(7.5² + 10.5²) = √(56.25 + 110.25) = √166.5 = 12.903 in. That is the distance the square advances along the board on each step-off.
- Cut stringer length. √(45² + 52.5²) = √(2,025 + 2,756.25) = √4,781.25 = 69.15 in.
- Throat. 11.25 − (7.5 × 10.5) ÷ 12.903 = 11.25 − 78.75 ÷ 12.903 = 11.25 − 6.103 = 5.147 in. That clears the 5 in DCA 6 minimum, with 0.15 in to spare.
- Bottom riser cut. 7.5 − 1.5 = 6.000 in for 2x treads.
- Angle. arctan(7.5 ÷ 10.5) = 35.54°.
- Stock. The layout plus end cuts needs about 81 in, so a 8 ft 2x12 yields one stringer.
- Stringer count. A 36 in stair at 16 in maximum spacing needs ceil(36 ÷ 16) + 1 = 4 stringers, which sets the real spacing at 12 in on centre.
Now check the arithmetic against the marks. Step 3's tread surface sits 3 × 7.5 = 22.5 in above the pad, and its riser face sits 2 × 10.5 = 21 in horizontally from the bottom corner. The diagonal from the bottom corner to the root of that notch is 2 × 12.903 = 25.81 in. Those three numbers are what the layout table on this page prints for every step, and they are how you check a stringer you have already marked before you commit a saw to it.
How to read the throat, the stock length and the stringer count
The throat is the pass/fail number. The American Wood Council's Prescriptive Residential Wood Deck Construction Guide (DCA 6) requires a cut stringer to keep at least 5 in of uncut depth, and it is written for 2x12 stock. That is why 2x12 is the default: a 2x10 at 9.25 in leaves only 3.4 in under a 7.5 in riser and a 10.5 in run, and is not a legitimate cut stringer at all. When the throat comes back under 5 in you have four honest options — widen the stock, reduce the unit run, reduce the riser (by adding a riser), or stop notching and switch to a solid stringer with cleats or proprietary steel tread brackets, which do not remove material at all.
The stock length shown adds 12 in to the layout for the bottom seat cut, the top plumb cut and saw waste, then rounds up to the next even foot, because framing lumber comes in 8, 10, 12, 14, 16, 18 and 20 ft lengths. A stringer over about 13 ft 6 in of diagonal — roughly a 9 ft rise — is at the edge of what a single 2x12 can span unsupported and normally wants an intermediate landing or a mid-span support post.
Stringer count is set by what the tread material can span, not by the stringer itself. Treat 16 in on centre as the working default for 5/4 radius-edge decking and 2x treads; many capped composite treads require 12 in on centre, and some require blocking under the nosing as well. The calculator places outer stringers at the edges of the stair and adds interior stringers until the clear spacing drops to your limit, which is why a 36 in stair at a 16 in limit comes back as four stringers at 12 in rather than three at 18 in.
Angle is a sanity check rather than a requirement. Anything from about 32° to 37° is a normal flight. If your angle is under 30° the stair is long and shallow; over 38° and it is steeper than the IRC allows for any combination of legal riser and tread.
Throat remaining in a 2x12 after notching
| Riser height | 10 in run | 10.5 in run | 11 in run |
|---|---|---|---|
| 6.75 in | 5.66 in | 5.57 in | 5.50 in |
| 7.00 in | 5.52 in | 5.43 in | 5.34 in |
| 7.25 in | 5.38 in | 5.28 in | 5.20 in |
| 7.50 in | 5.25 in | 5.15 in | 5.05 in |
| 7.75 in | 5.12 in | 5.01 in | 4.91 in |
The steepest riser and deepest tread the IRC allows together — 7.75 in on 11 in — is the only combination in this table that fails. Every column gets worse on narrower stock: a 2x10 at 9.25 in loses exactly 2 in from every figure here.
Cut the bottom, not the top
Two adjustments get confused. The bottom cut is shortened by the thickness of the tread material, so that the first finished step matches the others. The top of the stringer needs no such adjustment when it hangs off the rim joist at the deck surface, because the deck boards themselves form the last tread — but it does need adjusting when the stringer sits on top of a landing rather than against a rim.
Get the bottom drop wrong and every riser in the flight is right except the first, which is out by the full tread thickness. Both the IRC and the IBC limit the difference between the tallest and shortest riser in a flight to 3/8 in, so a 1.5 in error is four times the tolerance and cannot be argued away.
What goes wrong when cutting stringers
- Overcutting the notch corners. Running a circular saw past the layout line into the throat cuts the very fibres that carry the load. Stop the saw short and finish the corner with a handsaw or a jigsaw.
- Cutting all the stringers from the first one without checking it. Cut one, set it in place, and verify the top and bottom sit right before you use it as a pattern.
- Using a 2x10. It is 2 in narrower, which comes straight off the throat, and almost never leaves the 5 in DCA 6 requires.
- Forgetting the tread nosing. The notch is cut to the unit run; the tread board is the unit run plus the nosing projection and overhangs the riser face below.
- No solid bearing at the bottom. The stringers must land on a footing or a slab that will not settle, and be fastened against sliding. Size it with the deck footing calculator.
- Ignoring the top connection. A stringer hung on the rim joist needs a tested hanger or a ledger — toe-nailing a notched stringer to a rim is not a connection.
- Untreated lumber at grade. Exterior stringers must be pressure-treated to a ground-contact retention if the bottom is within 6 in of soil, and every field cut needs a brush-on preservative.
Where the stringer sits in the build sequence
Work in this order. Establish the rise and run first — the stair rise and run calculator settles the riser count and the unit run, and nothing here is meaningful until those are fixed. Lay out and cut the stringers second, using the numbers on this page. Then set the footing and the top connection, then the treads, then the guard.
For material takeoff, a stringer is priced by the board foot like any other framing member, and the board foot calculator converts your stock list into the units a lumberyard quotes. If the treads are decking rather than solid stock, the deck board calculator counts the boards and fasteners. The guard on an exterior stair over 30 in above grade is not optional; the baluster spacing calculator divides the rail so no 4 in sphere passes through the guard, and no 6 in sphere passes the triangle at the tread.
If you are pouring steps instead of framing them, the same rise and run drives a volume takeoff rather than a notch layout — see the concrete stairs calculator. And if your total rise is small and the run is generous, check whether a ramp is a better answer with the ramp slope calculator before you cut anything.
Key terms
- Cut stringer
- A stringer with the steps sawn out of the top edge so the treads bear directly on horizontal cuts. Also called a sawtooth or open stringer.
- Throat
- The narrowest remaining depth of a notched stringer, measured square to its bottom edge at a notch root. DCA 6 requires at least 5 in.
- Dropped bottom riser
- The shortened bottom cut of a stringer, reduced by the tread thickness so the first finished step matches the others.
- Step-off
- Marking the layout by walking a framing square along the board, set to the riser on one leg and the unit run on the other.
- Plumb cut and seat cut
- The vertical cut at the top of a stringer and the horizontal cut at its base — the two end cuts that fall outside the notched zone.