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Calculations you need to build stairs:
Here using simple arithmetic we explain how to make stair design calculations: number of steps, step riser height, total stair height or rise, total stair length or run.
We explain how total elevation change between two levels or floors (rise) and stair length (run) are used to calculate the right measurements when building indoor or exterior stairs to fit the building or the terrain.
Details of methods for accurate stairway rise & run measurement are provided for tough cases such as building a stair over steep slopes and irregular surfaces.
We describe how to translate the stair rise and run into a specific number of stair treads and risers that will be uniform and of proper (safe) dimension. We also describe how to design and build low-slope or low angle stairways with special consideration for tread and landing dimensions to avoid halting-walk stairs and other trip hazards.
This article includes example stair building calculations and warns about some "in between" stair tread sizes that may be a trip/fall hazard. We also explain now to include landings and platforms in stair design calculations.
Finally, we also explain how to adjust factory-built or pre-fab stairs to the exact stair rise dimension in your installation. Page top stair dimension sketch courtesy of Carson Dunlop Associates.
Our photo illustrates construction of a low-slope stairway located in the "Jewish Quarter" in Girona, Spain. These steps are several hundred years old, are worn, and have a bit of a slope to them.
[Click to enlarge any image]
Bernie Campbalik who taught us carpentry, including stair building, used a rule of thumb that basically makes the run longer when the rise is shorter.
I've seen several rules such as the sum of one tread and one riser should always be equal or greater than 17; or two treads plus one riser should add up to around 28 or 29.
The concept is that a low rise stair usually has, just as you suggest, a tread that provides a "longer" walking surface. Up to a point. If we make the rise too short (under 4 inches of rise) it's not a step at all, it's a trip hazard.
Generally we solve the problem of low slope long run stairways by using all platforms - steps that are 36" in length or more in the direction of run of the stairs, or by using a combination of normally sized stair treads (say 11" deep treads with a 6" or 7" rise) along with intermediate stair platforms.
And relevant to your question is the "depth" or run of each step in the direction of travel.
While generally it's good to use a deeper stair tread (treads less than 11-inches in depth are not recommended), there may be some intermediate depths (or step run) such as around two feet that make for awkward walking and may risk stair falls. A better stair design may involve increasing the rise and lengthening one or more steps into a platform of 36" or more of horizontal walking surface.
And of course long stair runs due to a very tall total rise (more than 12 feet) also is likely to require an intermediate stair platform as well.
Watch out: make these stair "steps" long enough to avoid a halting-walk stair fall hazard. See HALTING WALK STAIR DESIGNS for LOW SLOPES or SHORT STEP RISE
Definition of stairway rise: the rise of a stairway is the total vertical height of a stairway between the walking surface just before the first stairway tread or step up and the beginning of the horizontal walking surface reached at the top of the stairs.
Here is how you can measure the actual stair rise and run, followed by how to calculate the same data.
Our illustration above shows an example rise and run for a simple uniform-tread depth and riser height stairway. It's easy to measure the horizontal run and vertical rise for a stairway built between two parallel and flat surfaces - such as inside a building between two floors.
[Click to enlarge any image]
Simply extend a measuring tape down through the stairway opening in the upper floor and measure the distance between the surface of the upper floor or level and the surface of the lower floor or level. That will be the total rise.
But measuring the rise and run between two elevations that are not parallel and not flat, such as for an exterior stairway built over sloping ground can be more difficult (you cannot measure the vertical distance directly as we did above). If you're not careful you will make a measurement error, leading to miscalculations among total rise, run, and step and riser dimensions.
What many stair builders do is simply lay a (hopefully straight) stair stringer between the two elevations, measure the actual rise and run distances, and lay-out the stair dimensions on the fly by measuring and marking along the side surface of the stringer.
The approach of placing a stringer along the desired stair slope, measuring and marking layout immediately on the stringer works acceptably if terrain shape permits and if you are careful with stair layout.
The direct stairway layout marking along the stair stringer is easy if we are building interior stairs between floors (sketch at left) or exterior stairs down a concave slope (sketch below left) but this approach won't handle a convex slope or concave (below right) unless we first excavate the hillside or raise the entire stair design and assembly to span above the high point on the slope.
For laying out stairs whose top and bottom landings are themselves not level surfaces see STAIR DESIGNS for UNEVEN / SLOPED SURFACES.
For an easy way to measure the stair rise and run outdoors when the surface being climbed or descended (before your stairs are built) is uneven, see the sketch given just below at STAIR RUN MEASUREMENT: NUMBER STEPS
Definition of stair run or stairway run: the total horizontal distance that must be traveled in ascending the stairs between two levels.
Measure the total stair run. Indoors this is easy: Just make the measurement shown in our sketch above, measuring "Run" of the stairway as shown.
Watch out: don't just divide the horizontal run by your preferred tread depth (say 11") - instead you must calculate the number of steps in your stairway so as to assure that all of the steps have a uniform rise.
We'll do that next at STAIR RISE MEASUREMENT: NUMBER of RISERS
The total stair run distance will allow you to mark or stake out the exact location of both the stair top first step and the stair bottom last step. Outdoors you will probably need top and bottom stair landings unless you're entering and leaving the stairs from an existing sidewalk or ramp.
For direct measurement of a total stairway run and rise outdoors over uneven ground, use straight lumber of sufficient length as shown in our next sketch below.
Watch out: be sure that your boards are at right angles to one another and that the vertical and horizontal boards are plumb and level. Otherwise your measurements will be wrong.
Complete stair tread dimensions and specifications are at STAIR TREAD DIMENSIONS
We can and often do use this basic right triangle function a2 =b2 +c2 in building construction to be sure that we are placing all four sides at right angles to one another and/or to an existing structure by using the 6-8-10 rule to make sure, for example, that a stairway opening or a floor or deck is square or at right angles to a building wall.
Details about how to use the 6-8-10 rule to assure two framing members are at right angles to one another are at
For a different and interesting use of triangles and plane geometry to convert stair slope in degrees to tread depth and riser height
see CALCULATE STAIR TREAD DEPTH OR RISER HEIGHT FROM STAIRWAY SLOPE IN DEGREES.
Watch out: at STAIR RISER SPECIFICATIONS we emphasize that stair tread risers must be uniform in height, varying by no more than 3/8" from one step to another.
Greater variations in riser height are serious trip and fall hazards.
Also see FRAMING TRIANGLES & CALCULATIONS.
In our simple three tread four-riser stair sketch shown here you see the total rise is from lower walking surface to upper walking surface. Indoors that's floor to floor. Outdoors that's landing to landing.
Measure the total stair rise. Indoors this is easy: Just make the measurements shown in our sketch above, measuring "Rise" - the vertical distance of the stairway as shown.
Above we suggested by sketch how you can measure the stair rise over uneven ground. Again it's critical that your measurement boards are plumb and level and square.
Calculate the number of step risers: divide the total rise by a reasonable starting number for step riser height. We use 7 inches.
Watch out: don't forget to re-check check the riser height and actual number of steps against the horizontal run space to be sure your stair fits in the opening or space allotted.
And if your stair is indoors passing up through a cut-out opening in the floor above or outdoors passing through a multi-level deck or similar structure you will want to double check the headroom for your stairway too. See STAIR HEADROOM.
If your total rise is a multiple of 7 you're home free. That's unlikely. So now what?
Total rise measured R = 75"
First calculation: Rise in inches divided by 7-inches
Divide the rise total by step riser height: 75/7 = 10.71 risers. But we can't have "part of a step" - don't try that or everyone will fall down the stairs.
We will either increase the number of steps by 1, shortening the stair rise, or decrease the number of steps by one, lifting the stair rise.
Then we spread the adjustment evenly over all of the stairs,
I would calculate both numbers - that's going to be 11 risers or just 10 risers - and choose the one that keeps my step rise closest to a comfortable height.
11-riser stair example
75" total height / 11 stair risers = 6.81" risers individual stair riser height - this is an acceptable step riser height.
10-riser stair example
75" total height / 10 stair risers = 7.5" risers individual stair riser height - this riser is acceptable in some jurisdictions and not others.
I would build these stairs using eleven steps each having a 6.8" riser. I could try to make my first or last step 0.01" taller as that variation is much less than the allowable 3/8", but frankly you'll never sense or feel that 1/100th of an inch accuracy. Forget it.
If my riser calculation came up with a larger fractional step or riser I'd divide that amount by the number of risers to make an adjustment so that my step risers don't vary by more than 3/8".
96" total rise / 7" riser height = 13.7 risers.
I might not want 14 steps - perhaps there's not enough horizontal run space. I could build a 13 riser stair.
That 0.7 risers could be handled like this:
0.7-fractional riser x 7-inches per riser = 4.9 inches to be gained.
I will increase my standard step riser height of 7" by 4.9 (inches to gain) / 13 risers.
4.9" / 13 risers = 0.37" per riser that I need to add if I spread the adjustment exactly evenly over all 13 riser/steps.
Or I could round that figure to 0.4" giving me a 13-steps x (0.4 - 0.37) or a 0.39" error I have to make up over several treads.
That's because the maximum allowable variation in riser height is 3/8" or 0.375" - less than my 0.39" error.
I could thus split the 0.39 over two treads with a 0.19" or about 2/10" adjustment on each of two treads. That is within acceptable uniformity limits.
At STAIR RISER SPECIFICATIONS you will find complete details about stair riser specifications and codes, including the following excerpts:
Step riser specifications riser height (<= 7.75" or in some codes <= 7.0" or in Canada <=8.25") maximum and 4 inches (102 mm) minimum. This means your stair risers should be less than or equal to 7 3/4 inches. No taller. Some codes specify slightly taller stair risers, from 8" to 8 1/4" (Canada, for example).
Step riser height uniformity (<= 3/8" variation) - this means that more than 3/8 of an inch in variation of the height of steps from one step to another is a tripping hazard.
What is the run length needed for a rise of 108" using an 11" tread depth and 8" step rise - A.K. Debbie 11/24/2012
It's important to get comfortable with the basic math in stair calculations, and it's not too hard.
If we have to climb up 108" in height and we are going to make each step 8" tall (riser height is 8") we just divide 108 / 8 = 13.5 - so we'd need 13 1/2 steps - which is not quite acceptable as we need to end with an even number of steps. So I would adjust the rise until the numbers came out more nicely.
Since an 8-inch step riser height is a bit too high for safe comfortable stairway use anyway, instead of trying a still taller step to calculate the number of steps needed, I tried smaller stair rises.
I chose a sequence of numbers (7.5", 7.25", 7.125") dividing each of these into the total rise we need (108") to see how close to an exact number of steps we could achieve.
I noticed that if I set our riser height to 7.125" (about 7 1/8") we will use an even number of risers (15). Actually it's 15.1 but once we get our riser height close to exact we can make small adjustments of say 1/8" or less in the actual riser height for individual steps to make the stairway total rise come out exactly right.
Watch out: too much variation between individual step riser heights is a trip hazard. Most codes allow up to 1/4". I find that if we keep variation to 1/8" in step riser heights nobody will notice, nor be uncomfortable.
Now with 15 risers, if I make the individual tread depth (that's the horizontal tread run distance) 11 inches, our stairway will have a run of 15 x 11 = 165" in length.
I'd check to be sure that we had proper head clearance for the stairwell and that a run of 165" doesn't run into a horizontal distance or space problem before framing the stairs
Where space limitations require that your stairs make a single turn or where your stairs must ascend a height greater than permitted for a single run of stairs you'll need one or more stair platforms or intermediate landings.
For details of common building code specifications for stair landing dimensions and step riser and tread dimensions see STAIR PLATFORMS & LANDINGS, ENTRY
Is there an exception to the riser height variation for the very first step of the staircase? Let me attempt to clarify the question.
I have a deck (exterior porch) for which the distance from the top of the deck to the slab which forms the footing for the set of stairs is just shy of the 5 steps within a pre-fabricated 5 step stair stringer which can be purchased at a Home Depot or Lowes, for example.
If I attach the pre-made stringer from the deck to the slab, ensuring that the top of the deck to the next stair down is the same height as the rest, then the riser height from the slab to the first stair is greater than a 3/8' variation from the rest of the riser heights by 1/8th of an inch (ie. it's 1/2 inch shorter than the rest of the stairs -
I actually need to remove a half inch from the bottom most stair of the stringer to fit). If this is a violation of code, than it means I need to cut my own customer stringer. Just verifying. Any feedback is appreciated. - Dan
There is no exception for individual stair steps, first, bottom, top, or other. A difference in riser height can be a serious trip hazard at any location on a stairway.
Quoting from the article text above on stair and step height regularity and the amount of variation in stair step riser height that is allowed (presumably to avoid a trip hazard)
"Stair risers of uneven height - no variation greater than 0.375 inches is allowed"
As I read your note, you have just a 1/2" error to make up between the total elevation difference between the deck surface and the ground surface if you use a pre-fabricated stair way.
If you can split the adjustment between the top and bottom stair risers by trimming the stringer top and bottom, you'll have just 1/4" or 0.25" of riser height variation (one at stair top and one at stair bottom) - thus minimizing the trip hazard risk of the uneven risers and the variation will be within standards.
Watch out: be sure to measure the height difference (deck surface to ground surface) at a projected point along a horizontal line from the edge of the deck out to the location, in horizontal distance, of the front edge of the nose of the very first or lowest step of the stairway.
That will avoid any error in calculating total stairway height due to any slope in the actual ground surface.
If you need a greater adjustment in the stair height between the ground surface and the deck surface in order to avoid having to re-cut a whole new pair of stair stringers, sometimes that can be accommodated by changing the height of the surface of the concrete or other masonry platform that many building departments and local codes require be placed as a landing at the bottom of the stairs.
Low slope stair construction details are now at STAIR CONSTRUCTION, LOW ANGLE SLOPE
This explanation is now found at STAIR DESIGNS for UNEVEN / SLOPED SURFACES
This topic moved to STAIR DESIGNS, UNEVEN TOP & BOTTOM LANDINGS
We moved this subject to STAIR RISE & RUN GEOMETRY & CALCULATIONS
This has moved to STAIR CONSTRUCTION, LOW ANGLE % SLOPE
This discussion has been moved to HALTING WALK STAIR DESIGNS for LOW SLOPES or SHORT STEP RISE
This discussion has moved to STAIR DIMENSIONS, IDEAL
This discussion has moved to STAIR ANGLE & SLOPE
Also see STAIR DIMENSIONS, WIDTH, HEIGHT
This topic moved to STAIR CONSTRUCTION, SPECIFIC ANGLE or SLOPE
This discussion has been relocated to ADA STAIR & RAIL SPECIFICATIONS
In the United States stair specifications for climbable steps are discussed by the Americans with Disabilities Act, section 4.9, Stairs, from which we excerpt below.
Continue reading at STAIR DIMENSIONS, WIDTH, HEIGHT or select a topic from closely-related articles below, or see our complete INDEX to RELATED ARTICLES below.
Or see ROOF SLOPE CALCULATIONS for a complete explanation of the mathematics of calculating stair or roof slope, angle, pitch, rise, run, etc. as well as examples of using trigonometric functions such as tangent
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OSHA estimates that there are 24,882 injuries and as many as 36 fatalities per year due to falls from stairways and ladders used in construction. Nearly half of these injuries are serious enough to require time off the job--11,570 lost workday injuries and 13,312 non-lost workday injuries occur annually due to falls from stairways and ladders used in construction. These data demonstrate that work on and around ladders and stairways is hazardous. More importantly, they show that compliance with OSHA's requirements for the safe use of ladders and stairways could have prevented many of these injuries. -osha.gov/doc/outreachtraining/htmlfiles/stairlad.html