Author Robert Scharff, writing inWorkshop Math, explains and illustrates the following method of laying out stairs:
There are many different kinds of stairs, but all have two main parts in common: the treads people walk on and the stringers (also called strings, horses, and carriages) that support the treads. A very simple type of stairway, consisting only of stringers and treads, is shown in Figure 3-4, view A. Treads of the type shown here are called plank treads, and this simple type of stairway is called a cleat stairway, because cleats attached to the stringers support the treads.
A more finished type of stairway has the treads mounted on two or more sawtooth-edged stringers, and includes risers, as shown in Figure 3-4, view B.The stringers shown here are cut out of solid pieces of dimension lumber (usually 2-by-12), and are therefore called cutout or sawed stringers.
The first step in stairway layout is to determine the unit rise and unit run, shown in Figure 3-4, view B. The customary permissible unit rise for stairs is about 7 inches, but for a given stairway the exact unit rise is calculated on the basis of the total rise of the stairway.
The total rise is the vertical distance between the lower finish floor level and the upper finish floor level. This distance may be shown in the elevations; however, since the actual vertical distance as constructed may vary slightly from what it should have been, the distance should be measured. At the time the stairs are to be laid out in new construction, the subflooring is installed, but the finish flooring usually is not. If both the lower and the upper floor are to be covered with finish flooring of the same thickness, the measured vertical distance from lower subfloor surface to the upper subfloor surface will be the same as the eventual distance between the finish floor surfaces and therefore equal to the total rise of the stairway. However, if you are measuring up from a finish floor (such as a concrete basement floor), then you must add to the measured distance the thickness of the upper finish flooring to get the total rise of the stairway.
If the upper and lower finish floors will be of different thicknesses, then you must add the difference in thickness to the measured distance between subfloor surfaces to get the total rise of the stairway. Use a straight piece of lumber plumbed in the stair opening with a spirit level, or a plumb bob and cord, to measure the vertical distance.
Assume that a total rise measures 8'-11", as shown in Figure 3-5. Knowing this, determine the unit rise as follows. First, convert the total rise to inches. In this case it comes to 107". Next, divide the total rise by the average permissible unit rise, which is 7". The result, disregarding any fraction, is the number of risers the stairway will have. In this case it is 107/7, or 15. Now divide the total rise by the number of risers, in this case, 107/15, which comes to 7.13", or rounded off to the nearest 1/16", 7-1/8". This, then, is the unit rise.
The unit run is calculated on the basis of the unit rise, and a general architect's rule that the sum of the unit run and unit rise should be 17-1/2". If the unit rise is 7-1/8", the unit run is 17-1/2" minus 7-1/8", or 10-3/8".
You can now calculate the total run of the stairway. The total run is obviously equal to the product of the unit run times the total number of treads in the stairway. However, the total number of treads depends upon the manner in which the upper end of the stairway is anchored to the header. In Figure 3-6 three methods of anchoring the upper end of a stairway are shown. In Figure 3-6, view A, there is a complete tread at the top of the stairway. This means that the number of complete treads will be the same as the number of risers. For the stairway shown in Figure 3-5, there are 15 risers and 15 complete treads. Therefore, the total run of the stairway will be the product of the unit run times 15: 10-3/8" x 15 = 155-5/8", or 12'-11-5/8", as shown.
In Figure 3-6, view B, there is only part of a tread at the top of the stairway. If this method were used for the stairway shown in Figure 3-5, the number of complete treads would be 1 less than the number of risers, or 14. The total run of the stairway would be 14 x 10-3/8" plus the run of the partial tread at the top. If this run were 7", then the total run would be 14 x 10-3/8" + 7" =152-1/4", or 12'-8-1/4".
In Figure 3-6, view C, there is no tread at all at the top of the stairway; the upper finish flooring serves as the top tread. In this case the total number of complete treads is again 14, but since there is no additional partial tread, the total run of the stairway is 14 x 10-3/8" =145-1/4", or 12'-1-1/4".
When you have calculated the total run of the stairway, drop a plumb bob from the stairwell head to the floor below and measure off the total run from the plumb bob. This locates the anchoring point for the lower end of the stairway.
Cutout stringers for main stairways are usually made from 2-by-12 stock. First determine about how long a piece of stock you will need. Let us assume that you are to use the method of upper-end anchorage shown in the Figure 3-6, view A, to lay out a stringer for the stairway shown in Figure 3-5. This stairway has a total rise of 8'-11" and a total run of 12'-11-5/8". The stringer must be long enough to form the hypotenuse of a triangle with sides of those two lengths. For an approximate length estimate, call the sides 9' and 13' long. The length of the hypotenuse, then, will equal the square root of 9 squared + 13 squared, or about 15.8' , or about 15'-9-1/2". Figure 3-7 shows the layout at the lower end of the stringer. Set the framing square to the unit run on the tongue and the unit rise on the body, and draw the line AB. This line represents the bottom tread. Then draw AD perpendicular to AB, as long as a unit rise. This line represents the bottom riser in the stairway. Now, you have probably noticed that, up to this point, the thickness of a tread in the stairway has been ignored. This thickness is about to be accounted for by making an allowance in the height of this first riser, a process which is called dropping the stringer.
As you can see in Figure 3-4, the unit rise is measured from the top of one tread to the top of the next for all risers except the bottom one. For this one, the unit rise is measured from the finish floor surface to the surface of the first tread. If line AD were cut to the unit rise, the actual rise of the first step would be the sum of the unit rise plus the thickness of a tread. Therefore, the length of AD is shortened by the thickness of a tread, as shown in Figure 3-7, or by the thickness of a tread less the thickness of the finish flooring. The first is done if the stringer will rest on a finish floor, such as a concrete basement floor. The second is done if the stringer will rest on subflooring.