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Calculations you need to build stairs: here we start with simple arithmetic showing how to make stair design calculations. 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 2013 InspectAPedia.com, All Rights Reserved. Author Daniel Friedman.
Reply: Here is a simple approach to calculating the stair rise & run for a low slope stairway - starting rule of thumb
Great question. 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.
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.
Dan, Thanks for your response. I now think any rule can at best only make general recommendations for all the low-slope possibilities.
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.
Reply: stair design basics: calculating step riser height, step tread depth, total rise, total run, intermediate platform lengths
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.
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.
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.
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. 
But the rub is that we need to know at least two of the angles on this triangle. We know bc is always 90 degrees. 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.
Using the 6 8 10 rule in framing
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 aer 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.
Theoretical stair design before fitting to the actual data - how to calculate the number of uniform stair treads
Back to Tom's exterior low-slope stair design
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.
Add Steps & Platforms or Landings or use a Stairway of All Platforms to Avoid a "Halting Walk" Stairway
Find the Total Rise, Run, & Slope of Your Particular Stairway
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.
Check that the total horizontal run of the stairs fits the space
In the absence of a better term, we've coined halting walk stairs to describe a (usually low slope or shallow angle) stairway whose individual tread depth is deeper than common, too deep for passage by a single stride or step, but too shallow to provide a comfortable two or more stride or step progress up or down the stairway. Walking up or down a halting-walk stairway forces the user either to extend his/her stride to an uncomfortable length to reach the end of the tread, or s/he has to take an uncomfortably abbreviated final stride to get to the end of the tread. (See our little stick man taking halting-walk steps down the stairs in our sketch just below).
Either of these strides can result in an awkward placement of the foot, possibly not securely behind the tread nose, and in our opinion these stairways have an increased risk of trips and falls.
Why Halting Walk Stair Designs can be a Bit Awkward
Adding Stair Platforms to the Stair Calculation to Fix Halting-Walk Stairs
We need to make up (336-209 = 127) inches of run in the stairway. How do we get there while keeping stair riser height within standard recommendations?
Let's use some intermediate platforms instead of stair treads.
If we make the depth (in direction of travel) of each platform 36" we'd need 127 / 36 = 3.5 platforms. We have to solve this problem by finding a balance between the number of individual steps (at some width for which we started with 11" depth). We can make the platforms longer (deeper) in the direction of travel or we can make the stair treads deeper.
For this illustration we'll take the easy way out and make the platforms long enough (they are allowed to be longer than 36") to use up that missing 127" by simply dividing 127 by 3. (We don't go in the other direction, dividing 127 by 4 because then the platforms would be 31.5" = less than 36" in the direction of travel).
127" of additional run distance / 3 platforms = 42.3 inches in length (or "depth) in the direction of travel on the stairs.
So we need to change 3 of our 19 "steps" into platforms that are 42.3 + 11" = 53.3" in the direction of travel of the stairs. There are other ways to solve this problem depending on terrain, I've just picked this one. At an actual site we'd adjust the plan to best fit the site terrain and owner's aesthetic wishes.
Reducing the number of steps increases the individual rise for each of the remaining steps accordingly. For example if we eliminate three of our 4" rise steps then that (3 x 4"=12) 12-inches of rise has to be spread over the remaining (19 - 3 = 16) sixteen steps.
78" total rise / 16 steps = 4.875" riser height or if we eliminate 5 steps (19-5 = 14) we are back at 78" / 14 = 5/57" rise per step - which was your proposal. But if we keep the individual step tread depth at 11 inches then those intermediate platforms will have to get longer. I'll leave that calculation for the interested student to solve.
Final Stair Plan Calculations of rise & run & platform dimensions
To meet the rise and run you specified, in one option we can build a stair that includes
Or you could change the number of steps and platforms around to suit the terrain by making new calculations using my example.
If we make a "step" or more than one step if necessary, into a platform that is 36" long, we can comply even with strict code interpretations (since the shortest platform length in the direction of stair movement is 36").
An Actual Stair Design Fits the Number Stair Steps & Platforms & the Run or Depth of Each to the Total Rise & Run Data
Stair Design Tip: Actually if I were building these outdoor stairs I'd probably change the topmost step in our sketch below also a 4th platform, not adding any more rise but set into the ground at grade level to provide a stair entry platform - it really depends on what the stairs are rising "to" - grass, existing sidewalk, or what?
Stair Design Tip: We can make the top platform any length desirable because its surface has reached the upper level grade - extending it beyond that point does not count as extending the run of the stairs.
Stair Design Tip: We needed sixteen normal steps 11" deep x 4.1 rise, divided into three groups between platforms and landings. It's natural to think of groups of 5, 5, and six steps. But where should we put that extra or sixth step? I put it in the lowest set of steps in the plan (see sketch above) on the (OPINION) theory that longer runs of steps present a greater risk of falling - so let's put the greater risk of a fall closest to the end of the stairwell where the extent of a fall would be minimized. (f the stairs are properly designed and built this extra risk is probably too small to quantify.)
Watch out: As I know you realize since you asked the question, the trip and fall hazard to avoid is a sort of "mini step" that is just less than 4" in riser height but lacks sufficient run to avoid tripping. I figure if the natural stride of a person using such mini steps puts her foot landing right on the edge or nose of a step below (assuming descending), that's a serious fall hazard. Especially in the dark.
Your picture of the railroad tie staircase is similar to the staircase I have lived with for over 10 years and I agree with you on all the problems with this design. Now that the ties are rotting it is time for me to correct this.
You recommend a series of platforms as steps.
The question still remains: “Is there an established method to design the size of the steps?” I ask because I truly dislike the 36” deep platforms of my existing staircase. It simply feels unnatural to take a full-step and then a half-step on every tread. I believe the platform needs to be either longer than 36 inches (in order to take 2 full steps) or shorter; a longer platform doesn't’t work in my case. I’ve worn a path in the lawn alongside these awkward stairs, because that is actually the easier route.
But anyhow, that experience along with your last explanation gives me a better understanding of the “halting-walk” issue. I was hoping to find some established guidelines to go by but they all seem to cover conventional staircases only.
This staircase of mine is built outdoors at grade and being at a low-angle the usual staircase design rules don’t fit here. On the plus side, I am not too confined by the measurements as is often found indoors. I am no engineer but I believe the solution is to determine an ideal for the horizontal distance covered with one footstep; then use that dimension for the tread depth.
There are many possible ways to cut this up into individual steps; the question is what tread dimensions will allow me to use normal footsteps and avoid the awkward half-steps, especially after the rise is put into the mix. Personal preference will influence this but I have to consider all potential users.
Like you pointed out, the dimensions should allow the users to take consistent steps and not be forced to alter their stride by stretching out or cutting short in order to meet the next step in succession. I experimented with this, by drawing several series of lines on flat ground to indicate the stair nose at different intervals (20, 22, 24 inches etc). Then I paced over these marks and tried to determine which set felt the most natural to cross.
I realize the stride length of each user won’t be identical; even an individual will alter their stride length, depending on if they are traveling up / down or carrying a load.
Meanwhile I happened to find an existing staircase with 28” treads; after trying it out I knew my guesswork was getting close. I finally settled on a 27” tread depth. Then knowing the size and number of treads; it was possible to work out the riser height. This riser height works out to 5 5/8”.
Additionally the treads and center landing are sloped to promote drainage (1”rise in the length of each tread). Even with this slope added to the riser, the riser height is still 1 inch below the 7¾” maximum. This slope accumulates and reduces the overall number of treads and risers required.
You mentioned the most strict code interpretation limits the shortest platform length in the direction of stair movement to 36". That my steps could really be considered platforms is not a great concern to me. I’ve seen that code officials avoid giving design advice but will inform you when you are wrong. A “common-sense” design approach may be at odds with common-knowledge so I would appreciate any input on this. Perhaps there is someone out there who has built a hundred of these that could throw light on this subject.
Please see the accompanying drawing and comment. I’ve added more details because they all influence the design.
- Tom S. 9/29/2012
Once we make low slope (low angle) stairway treads deep enough (longer in your parlance) to avoid the halting walk problem I'm not sure there is a perfect tread depth for the reason you noted as well - horizontal stride varies enormously among individuals, especially when we add consideration for children, disabled or elderly (CDE my term) users.
But I think the issue of total length (depth) of your longer low angle stairway treads and the multiple strides that may have to shorten or stretch at the last horizontal stride is less of a worry than what I chose to call a halting walk stairway on which every step up or down is awkward. My reasoning is that once a stair tread has been made long enough, adjustment of the final stride to reach the end of the individual tread is not so awkward because the user has adequate total space to adjust his/her walking stride to a comfortable distance.
You will recall the 11 inches (279 mm) minimum for standard stairway tread depth.
We could theorize about just what constitutes an "undersized" or halting-walk stair tread depth for deeper tread but not deep-enough tread stairway. But because of the CDE problem I haven't figured out how to describe that minimum "too short" (or too shallow) trip-prone halting walk stair tread found on some low angle or low slope stairs whose tread depth already exceeds 11 inches.
Why use Intermediate Platforms to Extend Stairway Run
To my taste, a low slope stair on which every deep (long or say two or more stride distance) step makes me adjust my last stride by having to shorten or lengthen that stride can also be unpleasant if not as trippy as a halting-walk stair.
One can therefore build a stairway with multiple successive deep (long, multiple-stride) steps and treads (your thought-out design) or if it fits the required run distance, some may like my design of a stair with multiple runs of more standard individual treads (of more typical rise and depth) separated by one or more long (deep) platforms to add the total run length needed for a low slope stairway.
Your idea of using a platform to handle angles or turns is a very good practice that has the added feature of avoiding both the trippiness and the added construction trouble of angled steps and treads.
Together we invite comments and opinions from other stair builders.
Frequently Asked Questions (FAQs) about stair riser heights & height vs tread depth or run
Question: Using pre-fab stairs - Is there an exception to the riser height variation for the very first step of the staircase?
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
Reply: no stair step riser height variation greater than 0.375 inches is allowed
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)
How to Make Minor Adjustments to fit a factory-built stair to the specific overall stair rise
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.
Question: how do I calculate the stair run length for these measurements?
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
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.
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.
George we're back to plane geometry.
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, also see Use the 6-8-10 Rule
Questions & answers or comments about stair and step riser heights and dimensions and building codes.
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