How to Find Stair Stringer Length From Total Rise and Run

The stringer is the hypotenuse of the staircase. Its length is Pythagoras on a triangle whose vertical leg is the total rise, finished floor to finished floor, and whose horizontal leg is the total run. What catches people is that the two legs do not count the same number of things: the rise stacks up every riser, the run only counts the treads, and there is always one fewer tread than riser. Get the count right and the arithmetic is one line.

Risers set the count, treads set the run

Start with the total rise, measured from the finished surface of the lower floor to the finished surface of the upper floor. Divide by the riser height you are aiming for, somewhere near 7 inches for a comfortable domestic stair, and you get a number that is almost never whole. For a 108 inch rise at 7 inches you get 15.43. That is a choice, not an answer: 15 risers of 7.2 inches, or 16 risers of 6.75 inches. Both are legal, the 15 riser version is the shorter stair, and it is the one worked through below.

Whichever you pick, every riser is then the total rise divided by that count, and they are all identical. The run follows from the treads, and the tread count is where the stair stops behaving like a simple grid. Walk up a flight and the last thing you step onto is the upper floor itself. The top riser lifts you onto it, so it needs no tread of its own. Fifteen risers, fourteen treads, every time the stair lands flush with the floor above.

The stair triangle, dimensioned

The drawing puts the sawtooth against the triangle it lives on. Count the risers up the left leg and the treads along the bottom, and the mismatch that drives every stringer mistake is visible in one look.

A worked check you can repeat on paper

These are the numbers from the drawing. Every line is arithmetic you can do on a phone, and the last two lines are the same answer arrived at twice by different routes.

The second route never squares the rise. It turns the slope into a multiplier, 1.2630 inches of stringer for every inch of run, and multiplies. Two independent paths landing on 176.82 is the check that the riser count and the tread count were used in the right places.

Two diagonals, and they are not the same length

Here is a check that looks like it should work and does not. Take one step, treat it as its own little right triangle, and its diagonal is √(7.2² + 10²) = 12.322 inches. Fourteen of those laid nose to nose comes to 172.51 inches. The stair triangle said 176.82. The two disagree by 4.3 inches, which on a job is the difference between a board that fits and a board that does not.

Nothing is wrong with either number, they measure different things. Fourteen step diagonals climb 14 × 7.2 = 100.8 inches, and the floor above is at 108. That last riser is a vertical jump with no tread under it, so the step by step path stops 7.2 inches below where the stair actually ends. The stair triangle folds that riser into its vertical leg, which is why its diagonal is longer and why it is the one that sizes the board.

The practical consequence: lay the notches out with the framing square set to 7.2 and 10, and take the board length from the whole triangle. Then add material at both ends, because the plumb cut at the top and the seat cut at the bottom both sit outside the 176.82 inch layout. A 14 foot board is 168 inches and never reaches. A 16 foot board is 192 inches and leaves a working margin of about 15 inches for the cuts and the waste.

Mistakes that catch people out

• Counting one tread per riser. Fifteen treads at 10 inches gives a run of 150 and a stringer of 184.84 inches, 8 inches long. The stair also overshoots its floor opening, which is usually how the error gets discovered.
• Measuring the rise to the subfloor. If the upper level is getting three quarters of an inch of hardwood after the stair goes in, the true rise is three quarters of an inch more than the tape says. Cut before that is accounted for and the whole error lands in the top riser, which then stands proud of the rest.
• Letting the risers come out unequal. A rise that does not divide evenly has to have the remainder spread across every riser, not dumped on the bottom step. Codes allow only about three eighths of an inch between the tallest and the shortest riser in a flight, and the foot notices far less than that.
• Forgetting the bottom drop. The first tread sits on top of the stringer like all the others, so the bottom of the stringer has to be cut down by one tread thickness. Skip it and the first riser is a full tread taller than every riser above it.
• Buying to the calculated length. 176.82 inches is the layout diagonal, not the board. The seat cut and the plumb cut both need material beyond it, so round up to the next stock length rather than the next inch.

Try these

Common questions

How do you calculate stair stringer length?

Square the total rise, square the total run, add them and take the square root. A 108 inch rise with a 140 inch run gives √(11664 + 19600) = 176.8 inches. That is the line length along the slope, before the top and bottom cuts.

How many treads go with fifteen risers?

Fourteen, whenever the stair finishes flush with the floor above. The top riser lifts you onto the upper floor and that floor acts as the final tread, so the tread count is always one less than the riser count.

What is the maximum riser height?

Residential codes in North America generally cap the riser at 7.75 inches with a minimum tread of 10 inches, while commercial stairs are tighter at 7 inches and 11 inches. Risers in one flight must also stay within about three eighths of an inch of each other. Confirm the exact figures against the code edition your jurisdiction has adopted.

What is the bottom drop on a stringer?

The amount you cut off the foot of the stringer, equal to one tread thickness. Every tread sits on top of the stringer, so without the drop the first riser ends up one tread thickness taller than the rest.

Is the stringer length the same as the board length?

No. The 176.8 inch figure is the diagonal of the stair triangle. The board has to carry the plumb cut at the top and the seat cut at the bottom as well, so for this stair a 16 foot length is the practical buy and a 14 foot length is nowhere near enough.

How steep is a 7.2 by 10 stair?

About 37.6 degrees from horizontal, which is a 77 percent grade. Two risers plus a tread comes to 24.4 inches, inside the 24 to 25 inch band that most stair guidance treats as comfortable.

Related lessons and calculators

• Stair stringer calculator with code check, which runs the riser count, the run and the code limits for you.
• How to find rafter length from roof pitch and span, the same right triangle in roof language, run and rise instead of tread and riser.
• How to find the missing side of a right triangle, the bare Pythagoras that both of those calculations dress up.
• Grade to degrees converter, to turn the 37.6 degree stair slope into a percent or a ratio for a drawing.
• Pythagoras and slope calculator, for any rise and run pair with the working shown.
• the International Code Council, publisher of the residential code that sets the riser and tread limits quoted here.
• Area formulas for every common shape, a calculator covering fifteen shapes and an index of every geometry lesson here.