When tracing the layout of stair stringers, you have to consider many details if you want your steps to provide a comfortable stride while climbing or descending the staircase.
A fair amount of math is needed to figure out the rise and run—the height of its risers and the depth (or width) of its treads. Oversized components can strain your legs, while undersized components wear you out by shortening your stride.
The Blondel Concept
Stair construction is an old art form. Masons and carpenters have been perfecting the techniques for hundreds, even thousands of years.
A book published in 1675 by the French architect François Blondel under the title “Cours d’Architecture” describes a popular theory for calculating the treads and risers’ relationship for the perfect stairs.
In a section of this book, Blondel describes a theory based on the following assumptions:
1. The comfortable length of a step on the level is around 24-inches.
2. The height of the step to climb a vertical ladder is about 12-inches.
3. The natural length of a horizontal step is therefore double that of a vertical step.
4. Consequently, each part reduced in height is to be compensated by two parts of level (horizontal), and each part added to the height must be reduced by two parts of level so that in either case, both add up to a step of 24-inches.
For example, to make a step with a 4-in. rise, you will need to have 16-in. in width (horizontal depth) so that when the 16-in. of level is added to the 4-in. height (which is worth 8-in. in level), you’ll still have a sum of 24-in.
The same logic can be used for all different riser heights.
The 1st Method of Calculation
Without actually writing it as an equation, Blondel establishes in his work “Cours d’Architecture” that if you are going to go up as well as a long, then the horizontal travel distance should decrease to allow for the vertical rise with the conclusion that the horizontal length should be reduced by double the vertical rise.
It would seem also that versions of his theories in the conception of the perfect stairs were adopted in many countries worldwide. Blondel’s deductions can therefore be expressed in this formula:
Going + (Rise x 2) = 24 inches
With everybody’s physical makeup being different, there are certain considerations to consider when dealing with people either taller or shorter than the average.
A shorter step length may cause a shorter individual to trip, while a longer step length will force a tall person to shorten his/her normal stride—a stride being the distance starting from the point of initial contact of one foot to the point where the same foot lands again.
While conceiving a staircase for a specific individual, remember that their stride can certainly be affected by their height, age, illness, or injury. As you can see, it’s anything but an exact science that can easily become controversial.
A Modern Approach to Stair Calculations
Even though the International Residential Code (IRC) rules that the maximum height of each step (rise) should be 7-3/4 in. and the minimum depth (run) be 10-in., an 8-1/4 in. maximum rise, and 9-in. minimum run are pretty much the accepted sizing limitations that are generally used nowadays.
With any of those norms or practices, however, a tread of less than 11-in. must have an overhang of at least 3/4-in. to a maximum of 1-1/4 in., thus making the tread depth actually larger than the run.
After deciding on the dimensions of your rise and runs, trace it out on the stringer with a framing square to verify that there is still a minimum 5-in. width of stock left on the back (underside) of an opened stringer.
Stair building has evolved over the years to better accommodate the general population. Two methods are commonly used nowadays using slightly different formulas, as described in the second and third calculation methods below.
The 2nd Method
This equation adds the tread’s width (depth) to the riser’s height, resulting in a sum between 17 and 17-1/2 inches.
Going + Rise = between 17 and 17-1/2 inches
The 3rd Method
A more popular alternative, however, is a formula where the tread’s width multiplied by the riser’s height should result in a product between 72 and 75 inches.
Going x Rise = between 72 and 75 inches
There are, however, certain restrictions to be respected in both of these last equations, such as a maximum rise not exceeding 8-1/4 in. and a minimum run of at least 9-inches.
It should be noted that by IRC code, the maximum rise and minimum run could fall outside the boundaries in both of these last two methods.
Calculating a Staircase
The height of a staircase must be calculated from one finished floor to the next finished floor on the upper level, including subfloor, carpet, tiles, hardwood, parquette, laminates, or whatever other surface treatment will be further added to the floor.
The same logic also applies to the lower level floor. All further additions in floor treatment must be deducted from the overall staircase height.
A miscalculation of even 1/2-in. or so will throw the symmetry of the staircase out of sync, thus breaking your stride or causing a possible tripping hazard—the accuracy in its conception is proportional to the safety provided while ascending or descending to the next floor.
Step 1 - Measuring the Actual Height
Since the risers are limited by a maximum height when building a staircase, and any variances in the height of a riser directly affect the width of the run or the “going,” the first thing to do is to carefully determine how many risers will be required with respect to the 8-1/4 in. maximum width guideline.
This can be accomplished by first dividing the total height in inches from the standing floor up to the top surface of the next floor up by the maximum riser height of 8.5 in.
The best way to do this is to set up a stepladder directly under the stairwell opening and climb on it to hang a plumb bob from the landing edge of the opening, letting its weight hang down just shy of touching the surface so that you can mark the plumb spot on the floor beneath.
You can then measure the vertical distance from that spot right up to the surface of the floor above.
Just to clarify with a more concrete example, let’s assume that the height obtained from your measurement was 109-1/4 in.
Step 2 - Finding the Net Rise of the Staircase
At this point, you need to know if more materials such as an underlayment and a floor covering will be added to the upper floor.
As this will likely be the case—especially in most new constructions—an adjustment must be made to account for the combined extra thickness from those added layers, which are an integral part of the actual stair’s rise.
Let’s use an example where your building plans specify a 5/8-in. underlayment along with a 1/8-in. vinyl covering on the upper floor and having already measured the vertical height from the lower floor surface to the upper floor surface as being 109-1/4 inches.
Upper floor adjustment = 5/8” + 1/8” = 3/4”
Total elevation = 109-1/4” plus 3/4” = 110”
The total height with upper floor adj. = 110 inches
But since the lower floor is likely to be also on the bare subfloor and considering that all risers are to be at the same height (up to a maximum 8-1/2”), calculating the risers based on the elevation at 110-in. would cause the bottom riser to be less than all the other risers once the floor is finished.
Therefore you need to subtract the combined thicknesses of the underlayment and floor covering of the lower floor from the total elevation.
Using still the same example and assuming you’ll be using the same underlayment but this time covered with a 3/8-in. ceramic flooring secured with a 3/16-in. thinset, you’ll have to subtract 9/16-in. from the 110-in. total elevation.
Lower floor adjustment = 5/8” + 3/8” + 3/16” = 1-3/16”
The net rise = (the total elevation) minus (the lower floor adj.)
110” minus 1-3/16” = 108-13/16” or 108.8”
Note 1: When tracing the layout and cutting the stringers, this lower floor adjustment dimension will have to be added at step 8 (tracing out the bottom end of the stringer) for the wood to sit properly and provide consistent riser height throughout.
Step 3 - Determining the Number of Risers
From the dimension of the net rise of the staircase, you can now determine the total number of risers needed by dividing the net rise by the maximum riser height of 8-1/2.”
Number of risers = (Net staircase rise in decimal) divided by 8.5 in.
Therefore = 108.8” divided by 8.5” = 12.8 risers
Since the number of risers cannot have fractions or decimals, you must round off with the next higher number of 13 risers for this example.
Height for each riser = (net rise) divided by (number of risers)
108.8” divided by 13 risers = 8.37-in. or 8-3/8 in.
Step 4 - Calculating the Possible Tread Widths and Total Runs
Knowing that Going x Rise = between 72 and 75 in. and with the riser already established at a constant 8.37 inches, you can determine:
a) Minimum tread width = 72” divided by 8.37” (riser) = 8.6” or 8-5/8”
and b) Maximum tread width = 75” divided by 8.37” (riser) = 8.96” or 8-15/16”
The number of treads in any staircase is always one less than the number of risers, so in this example, you’ll have 12 treads.
a) Minimum staircase run = min. tread width x number of treads
8.6” x 12 = 103.2” or 103-1/8”.
b) Maximum run = max. tread width x 12 treads
8.96” x 12 treads = 107.52” or 107-1/2”
Note that according to the 17” - 17-1/2” rule, the maximum tread width in the example could be as wide as 9.13” or 9-1/8.”
Also, if you have plenty of room and would like to increase the total run of the staircase, you just have to add one more riser narrowing them to 7.77” or 7-3/4” high while stretching the total run between a minimum of 120-1/2” to 125-7/16”.
Tracing the Treads and Risers on the Stringer
Step 5 - Setting up the Framing Square
Once you’ve determined the rise and run dimensions of your steps, you’re ready to take your framing square and set it up to properly transfer those measurements to the stringer with a set of stair gauges.
A better way is to make a “framing square fence,” which will provide more support for your jig while tracing the top runner and the bottom riser.
You can simply make one from a piece of wood about 1-in. thick and 1-1/2 in. wide and about 24-in. long that you’ll partially rip down the center from both ends, leaving about 4-in. uncut in the middle between both saw kerfs (figure 1).
Each leg of the square can now be inserted into one of the slits. Slide the fence, so its inside edge (towards the heel of the square) lines up with the tread measurements on the outside edge of the body of the square (the larger arm), and with the riser measurement on the tongue of the square.
The fence can be secured to the square with two small clamps, and you have yourself a custom and reliable jig.
Step 6 - Tracing the Top Step
Placing the square on the top edge of the stringer with the tongue of the square (riser dimension) toward the top end, slide the square up for the point of its heel to line up with the end of the plank (figure 2).
Note 2: Before proceeding further, a crooked plank should be laid out with the crown up.
Also, make sure that the finished stringer will not have any knots, shake, or other major defects, but if you do have knots, as much as it is possible, move the layout over to keep the defect in the cutout zone.
Trace a line along the square’s blade (its larger arm) to mark the top tread of your stair on the stringer.
Perpendicular to that line, trace a line showing the trimmed end of the stringer by simply placing the jig against the underside edge of the stringer (figure 3).
This mitered cut is where it will eventually butt against the rim joist, the hanger board, or the stairwell joist. Keep in mind, though, that considering all the different ways to attach and fasten the stringers, the technique you’ll use can affect how to cut it.
Step 7 - Tracing the Subsequent Steps
Returning the jig on the top edge of the stringer, slide the jig down the stringer length until the tongue of the square intersects the line from the first tread (at the nosing).
Trace the location of the riser and the second tread, making sure at every step that the lines come together at their intersection point without crossing over each other.
A consistent layout is crucial, and the jig will provide just that as your aim should always be for accuracy and consistency.
Keep sliding the jig down the length of the stringer and marking as many steps as you need.
Step 8 - The Bottom End of the Stringer
Once all the treads are traced out onto the stringer, place the jig one step further down at the bottom of the stringer and trace the last rise and add one more line perpendicular to it to be used as a reference point with the finished floor line (figure 4).
At this point, trace every riser on the stringer at the same height without considering the treads’ thickness. Look at figure 4, where the bottom run and the two bottom risers are traced, and imagine the bottom tread at 1-1/4 or 1-1/2 in. thick resting on top of the bottom run.
It can easily be seen that the overall rise of the bottom step will increase by as much as that thickness, while the distance between the top tread and the finished top floor will be decreased by the same amount.
Adjusting for the Subfloor
If the finished floor is not yet installed, leaving the stringer sitting directly onto the subfloor, you’ll usually have to further adjust that bottom riser to ensure that the finished floor line eventually lines up perfectly with the floor covering.
This height was already calculated as the lower floor adjustment—refer to Calc. #2 in Step 2. Just use that result and add it to the “actual finished floor line” to create the stringer’s cutting line drawn parallel to the finished floor line.
Cutting Out the Steps
The steps should be cut using a portable circular saw, making sure that the shoe is perfectly perpendicular and the blade is very sharp, preferably carbide tipped, to provide a fine smooth cut.
As you cut the step out, make sure the blade stays clear of the other line perpendicular to it, and finish the cut with a handsaw or a reciprocal saw instead, so you don’t end up with an excessive amount of cut, weakening the overall strength of the plank.
With all these tips and recommendations, you can now proceed to build the perfect staircase, and you can follow the links below for more related information on the subject.