What is the angle of an American stairway?

it would have to be 45 degrees if the staircase is a right triangle.

no no penguin. The question is about the hypotenuse of a right triangle. And the slope can vary all the way to ninety degrees. Clearly the length and slope of the stairway is dependent on the length of the 2 legs meeting at the right angle. There is no “standard” angle.

The simplest way to find out would be to compare the height of the vertical part of a single step with the depth of the horizontal part. It seems to me, without doing the measurement yet, that the two lengths are about the the same, which would make it about 45 degrees.

@stanleybmanly There’s two bottom corners. One is 90 degrees and the other corner – the narrow end – is 45 degrees.

A stairway in the US must to be graded to accommodate the length of the average shoe. I don’t remember the exact details but I remember reading up all about this.

A few hundred years ago carpenters saved on the cost of construction materials by building steeper stairs. This meant the stairs could be steeper and still be safe. People then were smaller than they are now, thus their feet were shorter, but the stairs were still so steep that they were unsafe and many people were seriously injured due to falls.

Somewhere along the line somebody figured out that for safety, stairs needed to be wider and more gradual in incline. I’m sure it depends on the country but building standards today require stairs of a certain incline for safety.

@snowberry you mean the stair itself, not the stairway.

@janbb visualize a ramp leaning against a building. The building meets the ground at a right angle-90 degrees. If you move the foot of the ramp coser to the building, the slope of the ramp increases as doesthe angle of the ramp to the ground. The 2 angles (ramp/ground ramp/building) must add up to 90 degrees. Neither must be 45 degrees.

That depends entirely on the depth and height of the singular steps, and that can vary.
here it is between 41° and 20.72°

The height and depth of the steps must vary to accommodate the slope rather than the other way around. Think about ladders.

Preferred is 7 inch rise over an 11 run or about 32.5 degrees, here’s a Chart for stairs and fabrication. It runs from 24.8 degrees to 43.5 degrees.

I owned a house from 1720; the stairs down to the cellar were 45 degrees (stone blocks 10 inches on a side), front stairs were steeper than that about 50 degrees because of a short sideways hallway and back stairs & attic were about 38 to 39 degrees.

The angles are only 45°, if the sides creating the right angle are of equal length.

My stairs are longer than they are tall, so it cannot be 45°. What I mean is the length measured horizontally along the ground is longer than the height.

Yes, I get that now. So there is no standard angle.

Here is a write up from a company that specializes in concrete steps and rails in this area. Upstate Steps and Rails . From their brochure:

“Upstate Step and Rail offers over 100 various sizes and configurations of precast concrete steps. Sizes range in width from 3½, 4, 5, to 6 feet wide. The rise of each step is a constant at 7.25 inches
with our tallest step measuring 72.5 inches tall and being a 10 rise step. Each step in depth, is 11 inches.
All steps come with optional landings that are 11, 30, 42, or 60 inches in depth.”

So, they use a 7.25” rise to 11 ” run .
Tangent (angle) = 7.25/11 = .66 Arctan(0.660) = angle = ~33.5 degrees.

Sounds right, I eyeballed mine and it appears like a 30, 60, 90 triangle.

This is an interesting question. Here in New England we have some very old houses (for North America, that is) that have been occupied since being built over three hundred years ago when there was no such thing as a “building code”. So I’ve seen and walked my share of steep, narrow staircases. (And moved heavy furniture up and down them, too.)

But these days this is more heavily regulated. So I wanted to see what my own state of Connecticut says about that. And here’s what they say:

(Amd) 1009.3 Stair treads and risers. Stair riser heights shall be 7 inches maximum and 4 inches minimum. Stair tread depths shall be 11 inches minimum. The riser height shall be measured vertically between the leading edges of adjacent treads. The greatest riser height within any flight of stairs shall not exceed the smallest by more than 3/8 inch. The tread depth shall be measured horizontally between the vertical planes of the foremost projection of adjacent treads and at right angle to the tread’s leading edge. The greatest tread depth within any flight of stairs shall not exceed the smallest by 3/8 inch.

Read more: http://www.city-data.com/forum/connecticut/170753-stairs-construction-inspector-relocation.html#ixzz4vFr2sTpM

This is a general rule. There are exceptions for circular staircases, other types of dwelling units, retro-fitting stairwells in buildings whose earlier construction predated the Code, etc.

So the general rule is a rise-over-run of at most 7 ÷ 11 and at least 4 ÷ 11. I was wrong about that “at least”; clearly the grade can be much more gradual if the stair tread is deeper than 11”, and there seems to be no restriction on how deep that can be.

Apparently, you can also use your smartphone to measure the angle of your stair pitch, which seems pretty cool.

Here’s a site with alternating steps for staircases.

You want us to think of a scalene triangle, not a right angle one. Scalene has 3 sides of unequal length. The long side is the stairway and the angle opposite.the stairway is probably going to be greater than 90 degrees as you imagine it .and the sum of the other two will therefore be less than 90 degrees and the angle of the staircase to the floor will be less than 45 degrees.

The slope or rise to run is determined by the local building codes.

CWOTUS in his answer above talks about rise to run. It’s like the pitch on a roof

@Tropical_Willie Those stairs seem absolutely lethal. They take up twice the space and offer 3 times the trip hazards in so many ways. But (like high heels) they do look sexy.

Visited a Colonial House that had alternating steps up to the second floor bedroom, “start off on your left foot!”