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A staircase is a simple and elegant way for people to move between floors. The layout, design and execution by the builder all play a part in determining how safely and comfortably a stair accomplishes this task.
The horizontal distance from the front to back of the tread is called the run, and the vertical distance is the rise. As you will have noticed from the staircases you've walked on, the rise and run can vary.
Rise-to-run ratio. The total of the rise and run should add up to 17 inches. Practically speaking, this means that as the run gets shorter, the rise should get steeper for the stair to feel comfortable as you walk. There are limits, of course. A rise of more than 8¼ inches is going to be too steep for many people.
Consistency. Even more important, the ratio of rise to run must be consistent. If you stumble or trip on a staircase, it is probably due to a tread or riser that is not the same size as the others. Our bodies get in a rhythm when climbing or descending, and even an inch of difference can be enough to throw us off.
If you are thinking about adding flooring on top of existing flooring where it meets a staircase, keep this in mind. The treads may need to be adjusted as well.
Another thing that makes a stair comfortable to use is a tread overhang. Although some modern staircases omit this detail, an overhang will keep heels from scuffing risers on the way down and help you find your footing on the way up.
Another modern-style detail is to omit the riser entirely. You'll still have a rise, of course (you wouldn't get upstairs without it), but it's just empty space instead of a board. Building a stair this way means you won't be able to hide the supporting structure of the stair: the stringer. In the stair pictured here, the stringer is made of steel.
Most wooden staircases have two or three wooden stringers, angled lumber with sawtooth cuts where the treads and risers attach. For safety, a stair that doesn't have a riser should have a thicker tread, like the one shown. This reduces the space between treads.
Most codes will not permit an open riser unless the opening is 4 inches or less. Although you'll see stairs that don't follow this rule, a child could crawl through a larger space, so for safety's sake, build to code.
Ceiling height. The distance from the highest tread below a ceiling should be at least 80 inches, or 6 feet, 8 inches. As a 6-foot-3-inch man who lives in Philadelphia and spends a lot of time in historic homes, I can tell you that this rule wasn't regularly followed in the time before building codes. Don't make the same mistake.
On the way up, people will bump their heads if the ceiling is too low. On the way down, it's hard to judge ceiling height, so people will duck if it looks too close. Give people plenty of room whenever possible. A stair is a dangerous place for a fall, and the codes governing stairs are designed to prevent just that.
Although the tips above are not enough to teach you how to build a stair (that's one of the most complicated things a carpenter does), they will help you understand how and why they are laid out the way the are.