The total rise decides everything else
A flight of stairs has one input that is not negotiable: the vertical distance from the finished surface at the bottom to the finished surface at the top. Every riser in the flight has to be the same height, and they have to add up to exactly that distance, which means the riser height is whatever the total rise divided by a whole number happens to be. You do not choose 7-1/2 inches. You choose a riser count, and the arithmetic hands you 7.5789 or 7.4483 or whatever falls out.
Work an example. A total rise of 108 inches divided by a target of 7.5 gives 14.4, which rounds to 14 risers, and 108 divided by 14 is 7.7143 inches per riser. Thirteen treads at 10 inches gives 130 inches of total run, and the stringer along the notched edge is the square root of 108 squared plus 130 squared, which is 169.01 inches, or 14 feet 1 inch. That is the whole calculation. Fifteen risers instead would give 7.2 inches each and a 140 inch run, which is why the calculator shows you the neighbouring counts: swapping one riser moves the run by a whole tread depth and that often decides whether the stairs fit the space at all.
Note what the finished surfaces do to that measurement. If the bottom of the stairs lands on a slab that is going to get tile, the tile thickness is part of the total rise. If the top landing is bare subfloor now and will be 3/4 inch of hardwood later, that 3/4 inch belongs in the number too. Measuring rough-to-rough and cutting stringers off it is how a flight ends up with one odd riser at the bottom.
Treads, risers and the count that is one apart
In the usual arrangement the top of the stringer finishes flush with the landing, and the landing itself acts as the top tread. That gives one fewer tread than risers: fourteen risers, thirteen treads. Attach it the other way, with the stringer carrying its own tread at the landing level, and the counts match. Both get built, and the difference is one tread depth of total run, so it is worth being deliberate about which one you are cutting rather than discovering it when the stairs land a foot short.
Tread run and tread board width are also different numbers. The run is the horizontal distance from one riser face to the next, and that is what determines the total run and the stringer notch. A nosing that overhangs the riser below adds to the walking surface and to the board width you have to buy, but it does not move the next riser, so it never adds to the total run. A 10 inch run with a 1 inch nosing needs an 11 inch board and still occupies 10 inches of floor.
The rules of thumb, and what they are for
Carpentry has three old checks on whether a flight will feel right underfoot. They are proportion rules, developed from how people actually climb, and they long predate any building code.
| Check | Usual target | What it is getting at |
|---|---|---|
| Rise + run | 17 to 18 in | The simplest one. A comfortable step trades depth for height at roughly one for one. |
| 2 x rise + run | 24 to 25 in | The pace rule, from the length of a stride on the flat. The oldest of the three and the one most people trust. |
| Rise x run | 71 to 75 | Penalises extremes at both ends, so it catches a flight that passes the other two by being tall and deep at once. |
A 7.5 inch rise with a 10 inch run scores 17.5, 25 and 75, which is why that pair turns up so often as a starting point. A 7.7143 rise with a 10 inch run scores 17.71, 25.43 and 77.14, so it is slightly outside two of the three and it will feel a touch steeper than ideal. That is information, not a verdict. These are trade rules of thumb about comfort, and they are neither a legal test nor a substitute for one.
Where the code sits, and why this page will not quote it
Stairs are the most heavily regulated piece of ordinary carpentry there is, and for a good reason: they are where people fall. Maximum riser height, minimum tread depth, the permitted variation between the tallest and shortest riser in a single flight, minimum headroom, minimum width, nosing shape, handrail height and graspability, guard height and opening size are all set by the code your local building department has adopted. Those requirements are different for a private dwelling than for a commercial or public building, they differ between jurisdictions, and they change between code editions, so a figure that is correct in one place and year is wrong in another.
That is why this calculator gives you arithmetic and trade rules of thumb and stops there. Before you cut a stringer, ring or visit your local building department, tell them what you are building and where, and ask what applies. In most places a new flight of stairs needs a permit and gets inspected, and the inspector will measure the riser variation across the flight with a tape. Getting that conversation done before the stock is cut costs an afternoon; getting it after costs the stringers.
Notching, throat depth and the part that carries the load
A cut stringer is a board with triangles removed from one edge, and what carries the stairs is the strip of unbroken wood left behind the notches. Its width, measured square to the edge, is the throat. For a rise of 7.5 and a run of 10 on a 2x12 at 11.25 inches wide, the notch corner sits 6 inches in from the top edge, leaving 5.25 inches of throat. Five inches is the number carpenters generally treat as the floor, and steeper flights on the same stock eat into it fast: push the rise to 8 with a 9 inch run and the throat drops under 5.
When the geometry does not leave enough throat, the answers are wider stock, a doubled stringer, or abandoning the cut stringer entirely for a housed one, where the treads and risers sit in routed dadoes and the board stays full width. The last of those is how stairs were built before circular saws, and it is still the strongest option for an interior flight. As with everything else structural on stairs, whether a given stringer at a given span carries what it needs to carry is a load question, not an arithmetic one, and heavy or public stairs deserve a proper specification rather than a rule of thumb.
Questions people ask
How do I calculate the number of stairs from the total rise?
Divide the total rise by the riser height you would like and round to the nearest whole number, then divide the total rise by that whole number to get the riser height you are actually going to build. A 108 inch rise divided by a 7.5 inch target is 14.4, so 14 risers, and 108 over 14 is 7.7143 inches per riser. Never round the riser height itself and then multiply back, because the error piles up in the last step. The calculator also shows what happens at one riser more and one fewer, since a 15 riser version of the same flight is 7.2 inches per step and a foot longer on the floor.
What is a comfortable riser height and tread depth?
The trade rules of thumb are that rise plus run should land between 17 and 18 inches, that twice the rise plus the run should land between 24 and 25, and that rise times run should come out between 71 and 75. A 7.5 inch rise with a 10 inch run hits all three, which is why it is the pair most people start from. Steeper than that feels like a ladder, shallower feels like you are being made to shuffle. None of these are legal limits. Maximum riser and minimum tread are set by the code your building department has adopted and they are not the same everywhere, so check locally before you cut.
How long a board do I need for a stringer?
The stringer along the notched edge is the hypotenuse of the total rise and the total run: square each, add, take the root. A 108 inch rise with a 130 inch run gives 169 inches, which is 14 feet 1 inch, so you are buying 16 foot stock. Two things push it longer than that number. The cuts at the top and bottom are not square to the edge, so there is waste at both ends, and if the top hangs on a ledger or the bottom sits on a pad, that detail adds length. Buy the next size up and cut it to fit rather than working to the theoretical minimum.
Why is the bottom riser cut shorter than the others?
Because the stringer notches are cut to the riser height, but the bottom of the stringer sits on the floor while the first tread sits on top of the stringer. Without an adjustment, the first step would be a full riser plus the thickness of the tread, and every other step would be a plain riser. Taking the tread thickness off the bottom notch fixes it: with a 7.7143 riser and a 1 inch tread, the bottom notch is cut at 6.7143. The equivalent adjustment at the top depends on how the stringer meets the landing, which is what the landing selector on this page is about.
What is throat depth and why does it matter?
It is the width of solid wood left behind the notches, measured square to the edge of the stringer, and it is what actually carries the stairs. The formula is the board width minus the rise times the run divided by the hypotenuse of the two. A 7.5 rise and 10 run on an 11.25 inch 2x12 leaves 5.25 inches. Carpenters generally treat 5 inches as the practical minimum for a cut stringer, and steep flights on nominal 2x12 stock can fall below it without anything looking wrong. If yours does, use wider stock, double the stringers, or house the treads into a full-width board instead of notching it.
Do stairs need a permit?
In most places, yes, whether they are interior, exterior or a set of deck steps, and the same trip usually covers the handrail and guard that go with them. What is required, and how it is inspected, depends on your local building department and on whether the work is residential or commercial. The practical reason to ask before you build rather than after is that the two things inspectors check first on stairs, riser variation across the flight and handrail height, are both extremely awkward to fix once the stringers are cut and the treads are down.