The Equation of Stair Building

Staircases are not only a fundamental part of the architecture in multi-level homes, but one of the most important structural elements. Often stairs become just part of the background in life, used to get from place to place without any further thought, but architects and designers do not see stairs this way. They spend ample amounts of time designing them and craftsmen work diligently to install them according to design.

Working together to create such beautiful and grand structures, architects and craftsmen must use certain equations. These equations are to guarantee that the solution is always a striking and safe to traverse staircase.

Just when we thought we were done with Geometry…

Let’s take a look at all the math involved in order to get a perfect staircase.

For our example we will look at just a typical straight-run stair that simply goes from floor to floor in a straight line.

Start with the angle. The angle is very important and we must get it right. We will need to divide the rise, which is the distance from the first floor to the second, into equal parts. Each part cannot be more than 7 ¾ inches.

Now we will need to establish the overall length of the stair, called the total run. We will also need a space that is at least 36 inches wide, and we have to think about the angle. We don’t want our staircase to be too shallow or too deep. It needs to be comfortable to walk on.

The math continues with the treads and risers. These are the horizontal walking surfaces (treads) and vertical pieces at the back of each treads (risers). Staircases will vary in tread and riser length, but as with everything else, there are building codes that will require them to be within a certain range to ensure maximum comfort and safety.

Once all of these things are “added up”, stairs will start to be assembled which will only bring more equations.

First come the stringers. These are the saw tooth sides of the stair that the treads and risers will attach to. By taking a solid piece of wood and then cutting out the teeth, we'll have the basic structure of our stair.

Following the stringers are the treads and risers. Once these are installed, you'll have a finished stair that can be used to go from one level to the next.

For added safety, we will need to install handrails. Railings can't be more than 38 inches or less than 34 inches above a tread (or walking surface), and must be extended horizontally at the top and bottom.

And let’s not forget to provide at least 80 inches of headroom above any walking surface. But this is just the minimum. We may want a little more space if we’re extra tall.

Of course, there is a lot more math and hard work required to build stairs not mentioned here. This was just a quick overview to give you an idea of all of the hard work and mathematical aspects involved in building a staircase.

Article by Amber Burkhart