"You can absolutely cut a stair stringer if only you only knew the way the math works," my dad once said to me. So, long before I ever had a geometry class, I knew the math behind determining a slope.
It is really quite easy. A stair stringer is a solid 2-by-10 or 2-by-12 cut into a corrugated geometric shape that accepts a corresponding set of stair threads (horizontal members of a staircase) and stair risers (vertical members of a staircase).
A staircase that does not conform to the specific site dimensions is dangerous and violates all building codes, which is why you need to know your math.
In practice, stair stringers are cut according to the site dimensions of a specific project. Typically, we use the dimension between two finished landings.
There are several rules governing stair construction; two are pertinent to this issue. The first is that a stair rise should not exceed 8 inches (some municipalities allow an 8¼-inch maximum rise) or be less than 6 inches. The second essential rule is that all steps in a staircase (also known as a carriage) must be equal in height. If you consistently trip on a step, chances are that step is probably not the same height as the others.
Believe it or not, our feet develop an "anticipatory memory" of a stair's height; so much so that a slight variation ins step height will cause a trip or a stumble. Vocational teachers often demonstrate this phenomenon in carpentry classes.
EASY TO LAY OUT
It's easy enough to lay out your own stair stringers based on your specific land ing-to-landing dimensions. First, taking into consideration the thickness of your stair tread material, measure from the top of the finished landing above to the top of the finished landing below. This measurement is the total height of the staircase.
From this number determine a riser height (height of one single step) that will equally divide. For example: A landing-to-landing height of 60 inches will yield eight 7½-inch risers (60 divided by 8).
Once you have determined your specific riser height, using a carpenters square create an "L" that has a leg of 7½ inches and a leg that has the length of the stair's run, which is either 9 inches or 11½ inches depending on the type of staircase you are building. Lay this calculated "L" alongside the edge of a sturdy 2-by-10 or 2-by-12 and mark with a pencil consecutive steps so that the shorter leg of one "L" touches the longer leg of the next leg.
CHECK YOUR MATH
When you have worked your way down a piece of lumber, the stair stringer will start to look like a stringer. When you are finished with the layout, check your math and analyze the actual layout to make sure you have accounted for all the necessary steps. With a saw, cut the triangular sections from the 2-by making sure not to cut past the inside corner of the "Ls."
When complete, lay your stringer up against the landing and see how it fits. Accounting for the step's thread thickness, all your steps should be equal If everything looks correct, make a "sister" stringer for the adjoining side. A sister stringer is simply a duplicate of the first. if you have an extra-wide staircase, say, wider than 3 feet, you will probably need more than two stringers, so cut additional stringers as necessary.
Along with cutting a roof, cutting stairs is considered "higher mathematics" in carpentry. Sure enough, knowing the math means knowing some basics of geometry. Today it seems amazingly simple. Imagine how simple other things are that intimidate and keep us from learning?