Stair stringer calculator
A stringer is a right triangle with notches cut in it. The length is Pythagoras; the angle is arctangent; the only real decision is whether the board you own is deep enough to survive the notches.
Stair geometry
How this is calculated
- Stringer length = √(rise² + run²), measured along the top edge between the plumb cuts.
- Angle = arctan(rise ÷ run), in degrees from horizontal.
- Riser height = rise ÷ number of risers, where risers = ⌈rise ÷ 7.75⌉, because a 7.75″ riser is the maximum most residential codes allow.
- Remaining depth = board depth − riser height, measured vertically from the seat of the notch to the bottom edge. This is the material left under the notch, and it is the number that fails.
Worked example
A 36″ total rise with a 42″ run, cut from a 2×12 (11.25″ deep):
- Length = √(36² + 42²) = √(1296 + 1764) = √3060 = 55.32″
- Angle = arctan(36 ÷ 42) = 40.6°
- Risers = ⌈36 ÷ 7.75⌉ = 5, so riser height = 7.2″
- Remaining depth under the notch ≈ 4.1″ — thin, but above the 3.5″ that most carpenters treat as the floor
Push the rise to 42″ on the same run and the angle steepens to 45°, the remaining depth drops under 3″, and the 2×12 is no longer adequate. That is the point at which the answer is a landing, not a longer stringer.
The board depth is the constraint, not the length
Almost everyone calculates the length correctly and then buys the wrong board. A 2×12 gives 11.25″ of depth; cutting a 7.2″ riser and a 10″ tread out of it leaves a triangular notch that removes material from the weakest part of the span.
Two practical notes that do not show up in the arithmetic:
- Notch depth is limited by code. IRC R502.10 caps notching at one-quarter of the depth for solid sawn stringers — about 2.8″ on a 2×12. A layout that needs a deeper notch needs a different stair, not a deeper notch.
- Buy the next length up. A 55″ stringer comes out of an 8 ft board with 41″ of waste, which is fine if you need three stringers and the offcuts become blocking. It is not fine if you assumed a 6 ft board.
Where a stringer calculation stops helping
- It does not lay out the notches. That is a series of marks from a framing square, and the first one must account for the tread thickness.
- It does not check the stair code. Maximum riser, minimum tread, nosing projection and handrail height are in deck stair code.
- It does not size the stair for loads. Stringer count by width is a rule of thumb, not an engineered answer.
- It assumes a straight stair. Winders, curves and landings change both the geometry and the stringer count.
Questions people actually ask
How long is a stringer for a 36 inch rise?
With a 42-inch run, 55.3 inches between plumb cuts. That is the hypotenuse, not the board length you buy — allow for the plumb cuts themselves and buy an 8-foot board for three stringers.
What angle do deck stairs usually sit at?
Between 32 and 37 degrees for a comfortable residential stair. Above 40 degrees the stair feels steep and most codes will not accept the riser-to-tread relationship.
How many stringers do I need for a 36 inch wide stair?
Three. Two is acceptable up to about 30 inches wide; four is the usual answer above 48 inches, and some jurisdictions require them at 16-inch centres regardless of width.
Can I use a 2x10 instead of a 2x12 for stringers?
Only on a shallow stair. The notch removes depth, and a 2×10 at 9.25 inches leaves too little material under a 7-inch riser once a 10-inch tread is cut. A 2×12 is the standard for a reason.