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.
Green links show where you are. © Copyright 2015 InspectApedia.com, All Rights Reserved.
Perhaps one of the experts here could give advise on this. What are the rules for a low angle staircase? I am planning an exterior (built on grade) staircase on a slope too steep for a ramp.
My rise is 78 in. over 28 ft. and there is additional room for a landing at top and bottom. I am considering 3 ½” risers and 16” treads (using commonly available cement block).
What rules-of-thumb should I follow to build a comfortable walk-able low angle staircase? If there is some formula to the ideal cadence; I would form this in concrete in order to achieve it. I do not want to feel like I am taking “baby steps” or alternately taking a “step-and-a-half” all the way up and down. - Tom 6/23/12
Our photo (above left) 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.
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.
Our exterior stair photo at left illustrates one stair builder's approach to a low slope stairway. He used a combination of asphalt and landscape ties to build these nice looking steps that incorporate several trip hazards. The projection of the wooden tie up about 3/4" above the walking surface of each step threatens to catch the toe of a shoe, leading to a trip or fall.
And there is a naturally-occurring risk factor - wet surface of the steps. On these stairs the wet wooden landscape tie will be more slippery than the asphalt walking surface. Walkers will naturally tend to step on the wooden surface because it forms the leading edge of each step.
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
Dan, Thanks for your response. I now think any rule can at best only make general recommendations for all the low-slope possibilities. I figured that if I could determine the optimum number of steps to comfortably traverse the distance of the slope of my proposed staircase I would then know the correct number of treads.
So I marked off the landing locations and simply walked up and down the slope while counting the number of steps I took as I did so. I repeated this a few times adjusting my gait somewhat and arrived at an average number.
In this case I took 14 steps to travel the slope which equates to 14 treads - 24” long. Then since the landing counts as one of the treads; I divided the rise by 14 which give me a riser height of 5.57”.
I also counted steps made on level ground over the same distance as the length of the stair and came up with 12 so I think 14 is conservative considering the slope but about right for my needs. I am not in a hurry to build but before I do I would still like to hear a professional opinion or two on this. Another possibility might include a landing midway up thus making all the treads shorter? Your input is appreciated.
I think a sketch of what you're up to would be helpful - if you want to send one use the CONTACT US link at page top/bottom. Meanwhile I'll provide my own versions, not to scale, of your stair dimension proposal and an alternative that you might want to consider and then adjust to your taste.
Based on the figures you provided we describe for other readers the basics of stair design calculations that consider total rise, total run, and step riser height and step tread depth.
(June 14, 2015) Anonymous said:
i am ask to build steps for an older couple ,one already has a hip replacement an the other getting one done ,,, my steps height is three feet an the distance is five feet they need lower risers
can someone tell what the dimensions would be
You can make short rise steps by making the stairway run longer. For example if you have a total rise of five feet and you want a 3-inch step rise you'd divide (5x12=60) 60 inches by 3 inches to get 20 steps that you'd need to build to climb up five feet.
Of course you'll want to make the stair treads deeper front-to back to make such short rise steps safe and comfortable.
And secure hand railings on both sides of the stairs will be important for folks for whom stairs are difficult.
4 July 2015 bonita said:
I am no longer able to walk down stairs with the standard drop. How can I make my drop from step to step shorter?
When I'm no longer able to make it safely up and down stairs to my office I will probably install a chair lift - see STAIRWAY CHAIR LIFTS
The problem is this: you can indeed re-build a stairway to reduce the riser height but to do so the depth of each tread has to increase significantly too. That means that the entire run (horizontal length) of the stairway will get much longer than it is now, perhaps double.
In the interior of most homes, doubling the run of a stairway between floors would be horribly expensive - if it would fit at all, or making a long stairway that winds or turns - taking more room in other dimensions.
I suspect that the cost of a stair rebuild indoors would be much higher than the cost of adding a stair lift.
Definition of stairway run: the run of a stairway is the total horizontal distance traveled by a walker using the stairs, from the first riser to the last riser.
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 first illustration at left 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 (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.
However some terrain shapes are irregular enough or are humped so that you cannot just take off measurements directly. It's better to make an accurate and direct measurement of the stairway run and rise so that your calculated tread depths and riser heights will be proper and uniform.
Many builders use a transit to make accurate angle and distance measurements.
Certainly designing and building stairs with uniform tread depth and riser height over rough ground like that in our photo (left) can be a serious challenge.
Here we provide an alternative crude but effective approach to stair rise and run measurement using string, plumb line, and level, along with a stick or board (or two of them) of sufficient length.
If the horizontal run distance is small, you can get by with straight lumber tacked in place temporarily to permit measurement of horizontal run projection and vertical rise. Typically we use our stringer 2x lumber (before it has been cut to length) for this purpose since by definition the diagonal stringer will be longer than the horizontal stair run.
This approach works to measure and design a stairway over any irregular surface.
[Click to enlarge any image]
Make certain that the horizontal boards are straight and that they are held level and that your vertical board, useful for holding up the outer end of that upper horizontal member is also vertical.
Take off the rise and run measurements as we show in the sketch.
For longer distances than the length of boards you have on hand, you will do fine with string to set out the horizontal distance, a vertical pole, and a plumb line and measuring tape.
Below, for completeness but in small font, we include how to obtain stair rise and run lengths using sine and cosine tables - not a process we are seriously recommending.
Our photo shows a stairway during construction by the author & an associate (Arthur Cady). We faced a double difficulty of constructing a (replacement) exterior stairway over a rough hillside and stairs that had to land on a sloping surface.
By supporting the stringer above the hillside we bypassed the irregularities of the slope itself,but as you can see from the photo, because we had to land on a sloping surface the two stringers could not be identical, yet they had to line up properly for the treads to be level.
Using the longer (downhill) stringer as the mater pattern we measured and laid out the tread positions along the stringer surface, then aligned the two stringer tops and mapped the master layout onto the inner side of the uphill, shorter stringer. The treads were placed between the stringers, supported by both cleats and end-nailing through the stringers from outside.
Best design would have insisted on a level landing platform at the stair bottom but in the real world we can't always meet that criteria. In this case the stairs had to land on a narrow driveway.
Had we installed a level landing platform the drive would have become unusable by the client's vehicles. The result is a first step with uneven riser height depending on where you enter the stairway.
The geometry approach to calculating stair rise and run over rough terrain when you have only two measurements works correctly provided you know enough triangular geometry to convert between your sloped-line measurement (the hypotenuse of the triangle) and the triangle's other sides that form the rise & run dimensions. Using a calculator and it's square root function, and the most basic knowledge of geometry is not enough - we also need a table of sine and cosine values, functions readily found these days online. 
a2 =b2 +c2 - the square of the length of the hypotenuse (a) equals the squares of the lengths of the opposite sides of a right triangle (b) and (c).
In a geometry or trigonometry text you'll see that when we refer to the angle ab we mean the angle formed at the intersection of lines a and b, or in this case the lower left angle in our sketch.
And so the three angles formed by the triangle's sides are ab, bc, and ac.
Knowing that the triangle abc will always include a 90 degree right angle ( i.e. this is always a "right triangle"- angle bc in our sketch) allows use of sine and cosine tables to obtain the lengths of the two unknown sides b and c in the sketch.
Geometry teaches that if we know three pieces of data about a right triangle (one side length and two angles, or two side lengths and one angle) we can obtain all of the other data about a triangle - all of its angles, and the actual lengths of all of its sides.
sin of angle ab = (length of the opposite side c / the triangle's hypotenuse a) = a / c
cos of angle ab = (length of the adjacent side b / the triangle's hypotenuse a) = b / a
On the ground we'd have to measure either angle ab or angle ac. You can do this using a transit, a protractor or other angle measuring tool either by placing your angular measuring tool on the sloped surface of the stringer side a, or by actually measuring the angle formed between side a and a horizontal or level surface.
Our carpenter's square at left is one such tool, though there are some neat level tools that, placed on a slope, will simply give you the angle directly as a readout.
Example: if our stairs are to run from the two extreme ends of the 2x stringer we illustrate at above right, and if that length (after cutting to "fit" the hill and desired stair run) measured 100 inches exactly, using just that known length of the hypotenuse of the triangle we would obtain the key measurements of stair rise c and stairway run b by first converting all measurements to inches and then using a table of sine and cosine values. Suppose we measure angle ab (sketch above) at 30 degrees.
sin of angle ab = (length of the opposite side c / the triangle's hypotenuse a) = a / c = 100 / c
cos of angle ab = (length of the adjacent side b / the triangle's hypotenuse a) = b / a = b / 100
Reading from sine and cosine tables, we'll find that side c (rise) = 50" and that side b (run) = 86.6"
If you don't think this whole sine-cosine process is absolutely horrible for field use in building a set of stairs then we're not on the same planet.
We can and often do use this basic right triangle function a2 =b2 +c2 in deck building to be sure that we are placing the deck sides at right angles to the building by using the 6-8-10 rule to make sure that 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 FRAMING TRIANGLES & CALCULATIONS.
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.
A consistent riser height of 5.57" would be entirely in spec if you can achieve that for all of the risers, presuming we make all of the steps identical in tread depth - which is SOP. But the landing, if its walking surface is at ground level, doesn't count as a riser. It's a step or tread (really a platform) but not a riser. Let's compare your 14 steps of 5.57" rise and 24" run to your data.
For details of common building code specifications for allowable stair landing dimensions and step riser and tread dimensions see:
Use your browser's back button to return here.
Stair codes talk about slope chiefly when discussing how much out of level a stair tread may be from front to rear or from side to side to avoid a slip and fall hazard.
But the maximum stair slope for the overall stairway for stairs used as a public passageway between levels is also implicit in the maximum step riser height - typically 8" or in some codes such as New York, 8.25" maximum riser height.
I'd like to propose an alternative design that increases the number of stair steps, uses a smaller rise, and adds some longer walking platforms to avoid what I call those "halting walk stairs" that we often see. I must emphasize that this is my opinion. Your local building official has the final word on accepting your stair design.
Re-stating the problem from a stair design basics view, when we are going to custom-build stairs to carry people between two elevations, we divide the total rise from level 1 up to level 2 by a number of risers that will give us a standard riser height that is within standards - preferably between 4" and 7" of rise.
In our photo you can see an implementation of a stairway on a very low slope rise at the Mohammed V. Mosque in Casablanca, Morocco. The builders made each step a long walking surface, more than 36-inches in this case, and they also provided a visual cue of the location of each step by using a light colored stone riser.
In these Moroccan stairs the risers were uniform in height and each riser was very close to 10 cm (about 4" U.S.).
Using Tom's data:
If we used the minimum rise/step of 4" then 78 / 4 = 19.5 - we'd have to go to 19 steps (since we can't build a fraction of a step and since we're not going with 20 steps which would have a slightly short rise of 3.9" though the local code official might accept that)
Really this means we need 19 steps up or 19 risers to make up the vertical rise - the total change in grade.
Watch out: we still have to check the riser height and actual number of steps against the horizontal run space as follows.
For most indoor and many outdoor steps and stairs we simply choose a standard tread width (say 11") and we allow the number of steps to determine the total horizontal run of the stair case. Then we "fit" that stair into the building design plan, or outdoors we let the stairs end where they may, provided there is adequate entry/exit room at the stair top and bottom.
But for our example low-slope outdoor stair situation, the stairs have to climb a low slope, traversing extra distance. Here is how we solved that problem.
Let's use an 11-inch deep tread to make nice stairs. Actually I don't like an 11-inch deep tread with such a short rise but it's code compliant. [Click any of our images to see an enlarged view.]
The total rise needed for the stairs is in conflict with the total run needed for the stairs.
In other words, you can't keep stair riser height within an acceptable range and traverse the horizontal run distance without deepening the stair treads.
This means you're going to make your stair treads deeper, but unfortunately, keeping riser height to within acceptable bounds and making the stair run match the required length, the result may be an awkward halting-walk stairway.
This discussion has been moved to HALTING WALK STAIR DESIGNS for LOW SLOPES or SHORT STEP RISE
Low or short rise deep or "long" stair tread designs that descend a long gentle slope and indoor stairs being designed for seniors, children, or others who may have difficulty ascending or descending steps
Also see STAIR DESIGN for SENIORS
At left there is a trip hazard (a toe catcher) on these outdoor steps because the landscape ties extend up above the black asphalt walking surface near the end of the stair step. But there is a second trip hazard: the total step depth from nose to riser may be too short for comfortable walking.
Watch out: in reducing the step riser height if you produce a tread depth distance that makes walking awkward or "halting" the result may be an increase risk of stair falls.
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.
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.
George we're back to plane geometry.
[Click to enlarge any image]
You are asking the rise and "run" of the treads. There is no single answer, since we could choose different tread depths or "runs" that would give different tread rises or heights. E.g. we could make the stairs "one giant step" or "three little steps".
But to stay within reason we chose a stair tread depth (step run length) of 10" in the equation above to obtain an individual step rise of 7.8". That is, on a 38 degree sloped stairway, we will ascend 7.8" in height for every 10 inches of horizontal distance traveled.
Complete details about converting slope or angle to stair rise & run along with other neat framing and building tricks using triangles and geometry are found
at FRAMING TRIANGLES & CALCULATIONS. And for a special use of right triangles to square up building framing,
see Use the 6-8-10 Rule detailed explanation and examples for framing squarely and at right angles
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. You will see that the ADA section 4.9 "Stairs" does not explicitly discuss short or low-riser stair steps nor "easily-climbable stairs" but instead is focused on step and railing standards to avoid falling hazards on normal stair dimensions.
Also see our discussion of stair headroom and ADA 307.2 on protruding objects at STAIR HEADROOM
Continue reading at STAIR RISER SPECIFICATIONS or select a topic from the More Reading links or topic ARTICLE INDEX shown 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
Green link shows where you are in this article series.
OR use the Search Box found below at Ask a Question or Search InspectApedia
Try the search box below or CONTACT US by email if you cannot find the answer you need at InspectApedia.
Please see the stair calcualtion discussions, questions, replies at STAIR RISE & RUN CALCULATION FAQs
Use the "Click to Show or Hide FAQs" link just above to see recently-posted questions, comments, replies, try the search box just below, or if you prefer, post a question or comment in the Comments box below and we will respond promptly.
Search the InspectApedia website
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