Stair Calculator

Nobody remembers the thirteen steps they climbed without thinking. They remember the fourteenth, the one that was a little taller than the rest and caught a toe.

A stair makes a promise with its first two steps. Researchers at the National Bureau of Standards described the mechanism in a 1979 guide written for the Consumer Product Safety Commission. Once the first tread feels the way it looked, and the second confirms it, the climber takes the stair to be uniform. He shortens his stride, trims the allowance he leaves to clear each riser, and is free to look around. From then on the body is running a pattern. A step that breaks the pattern is met by a foot that was aimed somewhere else.

That is why stair building is mostly division. The building supplies one stubborn number, the height from one finished floor to the next. The carpenter’s job is to cut it into equal parts and then to defend that equality through every board that follows.

The foot stops looking

The effect has been measured. In a 2021 study in PLOS ONE, twelve young adults climbed a seven-step laboratory stair on which one riser, the fifth, had been made 1 cm taller than the others. Nobody adjusted. Foot clearance over the odd step fell by 0.8 cm compared with the even stair, and the authors’ explanation is that the inconsistency goes unnoticed.

A larger experiment, reported in Applied Ergonomics a year earlier, found the same thing in 26 younger and 33 older adults. Going up, both groups cleared the taller step by about 9 mm less. Coming down, they landed with 3 percent less of the foot on the tread. The participants did not change the path of their feet, which means they either failed to detect the difference or could not adapt to it. A toe that clears by less can catch. A foot with less tread under it can slip past the edge. The authors’ conclusion is one that a framing crew could pin to the wall: consistent stair construction is vital.

The scale is not small. An analysis of emergency department records estimated that 24,760,843 people were treated for stair-related injuries in the United States from 1990 through 2012, an average of 1,076,558 a year, and two thirds of those visits were by people between 11 and 60. The Bureau of Standards guide had put the traffic in perspective: 1,953 billion uses of stair flights in 1975 against 264 million noticeable missteps, roughly one in 7,400. Most missteps end in nothing. The few that do not add up.

Irregular stairs were ordinary in the homes that guide drew on. Investigators who went out and measured found steps that differed from their neighbors by an inch or more on 46 percent of the stairways in their sample.

Dividing the height

The division itself is short. Measure the total rise from finished floor to finished floor, as Andy Engel instructs in Fine Homebuilding, allowing for any flooring that has not been laid yet. Divide by the riser you would like and take the nearest whole number. Then divide the rise again, this time by that number. The first division says how many risers there are. The second says how tall each one really is, and the second figure is rarely friendly.

Take a floor-to-floor height of 108 inches and a target riser of 7 1/2. The quotient is 14.4, so the stair gets 14 risers, each 108 divided by 14, or 7.714 inches: a hair over 7 11/16. There are 13 treads, because, as Engel notes, there is always one less tread than there are risers. The upper floor is the last step. At a 10 inch run the stair covers 130 inches of floor and climbs at 37.6 degrees. The calculator above opens on exactly this stair, and every figure in the paragraphs that follow can be read off it.

The decimal matters. Fourteen risers of 7 11/16 inches come to 107 5/8, which is 3/8 inch short of the floor above. A builder who steps off the rounded figure fourteen times hands the whole error to the top step. The cure is to measure every mark from one end, so that each is rounded once and nothing is carried forward. On the board, a framing square set to one riser and one run advances 12 5/8 inches along the edge per step, and the last mark lands 176 13/16 inches from the starting one.

Whether a riser and a run feel right together is an older question. Nicolas-François Blondel, in the last volume of his Cours d’architecture, published between 1675 and 1683, was the first person known to tie the two to the length of a stride: twice the riser plus the tread equals one step. English building guidance still states the relationship, and puts twice the rise plus the going between 550 and 700 millimeters. The example stair comes to 646.

What the saw takes away

A cut stringer is a board with a row of triangles removed. Each notch is drawn with the square, one leg set to the riser and the other to the run, and each takes a bite out of the board’s width equal to the riser times the run divided by the diagonal between them. For the example that is 6.11 inches. What is left underneath, measured square to the edge, is the throat. The throat is all that carries the stair.

This is why the 2x12 is the stringer board. Its dressed width, in the lumber size table of the Forest Products Laboratory’s Wood Handbook, is 11 1/4 inches, which leaves the example a throat of about 5 1/8. A 2x10, at 9 1/4, would leave 3 1/8. The American Wood Council’s deck guide says all stringers shall be a minimum of 2x12 and marks a throat of 5 inches minimum on its stringer drawing. Its commentary adds that cut stringers were analyzed at a depth of 5.1 inches, the figure for a 7 3/4 inch rise on a 10 inch run, which is the steepest stair the residential code permits.

Notching costs strength, and the same guide shows how much. It holds a cut stringer to a span of 6 feet against 13 feet 3 inches for a solid one, and asks for at least three of them, no more than 18 inches apart. Roe Osborn, writing in Fine Homebuilding, supplies the craft rule that follows from it. Stop the circular saw at the line and finish with a handsaw, because overcutting the stringer weakens it.

Then comes the adjustment that separates a stair that works from one that trips its own builder. The square lays out every riser the same, but the finished stair is not built the same way at both ends. Every tread adds its thickness on top of a notch. Nothing is added under the foot of the stringer. Left alone, the first step would be one tread thickness too tall and the last would be that much too short. So the bottom of the stringer is cut off by exactly that thickness. With 1 inch treads, the first riser cut of the example is 6 11/16 inches, and every finished step is 7 11/16.

Deck builders call it dropping the stringer. Forgetting it is the classic way to build a crooked stair from perfect arithmetic.

The numbers in the code

Building codes turn all of this into limits. The 2021 International Residential Code, the model code published by the International Code Council, caps the riser at 7 3/4 inches and sets the tread depth, measured from nosing to nosing, at no less than 10. The nosing must project between 3/4 inch and 1 1/4 inches unless the tread is at least 11 inches deep. A stairway must be at least 36 inches wide. A single flight may not rise more than 12 feet 7 inches between floors or landings.

And it writes the uniformity down. Within any flight the tallest riser may not exceed the shortest by more than 3/8 inch, and the same tolerance applies to tread depths, all measured without carpets, rugs or runners. That allowance is for wood and workmanship. It is not a budget for arithmetic. When the Bureau of Standards surveyed the model codes of the 1970s, some allowed 3/16 inch and at least one required no particular uniformity at all.

Headroom is the limit that reaches back into the framing. The code asks for 6 feet 8 inches, measured plumb from the sloped line that joins the tread nosings, so the low point is wherever a ceiling or a beam ends above the climbing stair. On the example, under a ceiling 97 inches above the lower floor, an opening 120 inches long leaves 6 feet 8 13/16 inches over the second step. Shorten the opening by two inches and it fails. The length of a stairwell is a headroom calculation before it is anything else.

None of these numbers is a law of nature. Minnesota’s rules repeat the 36 inch width and the 6 feet 8 inches of headroom, while Pennsylvania adopts the same model code but keeps an 8 1/4 inch riser and a 9 inch tread. The federal guide to accessible design holds risers between 4 and 7 inches, with treads of 11 or more. Workplace rules allow a standard stair anywhere from 30 to 50 degrees, with risers up to 9.5 inches. England lets a private stair rise 220 millimeters a step at a pitch of up to 42 degrees. What every one of them shares is the demand that the steps of a flight match.

What a layout cannot settle

A straight stair is the only stair this arithmetic describes. Winders, landings, turns and spirals have their own geometry and their own code sections, and none of it is here. Nor is the structure. The throat is a geometric fact, not a strength rating. Whether three notched boards will carry a given stair depends on span, species, grade and fastening, which is why the deck guide prints span limits, and why some builders have moved to engineered lumber. Mike Guertin describes laminated strand lumber stringers 14 to 16 inches deep that are straight and do not shrink.

Sawn lumber does shrink, Osborn warns, most of all around the notches, where end grain is exposed. Boards crown, and Engel lays his stringers out with the crown up. Finish flooring that arrives after the stair changes the first and last riser by its thickness. A calculation assumes none of that.

A passing check is not an approval either. The limits quoted here are one edition of a model code. The stair in front of you answers to whatever the local authority adopted, and to the inspector who reads it. Guards and handrails are not checked at all.

What the arithmetic can do is keep the promise made by the first two steps. Divide the height once. Carry the decimal. Drop the first cut. Leave 5 inches of wood under the notches. A stair built that way gives the foot nothing to notice, and unnoticed is what a stair should be.