Stair building to a specific percent slope or angle:

Here using simple arithmetic we explain how build stairs of a desired slope, using the comfortable angle of a 38° stair.

We describe how to translate the stair rise and run into a specific number of stair treads and risers that will be uniform and of proper (safe) dimension. We also describe how to design and build low-slope or low angle stairways with special consideration for tread and landing dimensions to avoid halting-walk stairs and other trip hazards.

This article series includes example stair building calculations and procedures. Page top stair dimension sketch courtesy of Carson Dunlop Associates, a Toronto home inspection, education & report writing tool company [ carsondunlop.com ].

How to Build a Stair to a Specific Angle - 38°

How to Convert a Stair Angle in Degrees to Stair Rise & Run Dimensions

38° as a stair angle is very close to our ideal given at ANGLE & SLOPE of an IDEAL STAIRWAY and also in our table of COMMON RAMP, STAIR & LADDER DIMENSIONS, ANGLES, SLOPES

[Click to enlarge any image]

How the heck would you build a 38° stair? One way would be to lay your protractor or angle-level on a stinger and start making on-the-ground measurements.

But it's pretty easy to just do the calculations and you'll know exactly what step rise and run you need to get the stair angle you need for a stairway of any total rise or total run length.

Really? No. You may have to adjust the step rise slightly to fit the exact run length but that's pretty easy to do after we have our stair angle calculations.

At the end of this discussion I'll give links to "how to" articles to make the stairs you build fit the actual situation or building involved.

Using our sketch at left, we want to express the angle of the stair (it's slope or rise over its run) marked as "38 deg" at the left end of our stair trangle.

The stair angle or "angle T" is defined as Vertical Rise divided by Horizontal Run.

Our client said they want the stair angle to be 38°.

My trigonometry teacher taught me that Tan or Tangent is a ratio of the two sides of our triangle - the sides opposite to and adjacent to the angle formed by the horizontal run X₁, the vertical rise Y₁, and the slope or hypotenuse of the triangle those form.

The Tangent of any angle θ (in this case 38°) is a ratio of the angle's opposite side (vertical rise in our case) and its adjacent side (horizontal run in our case).

The Tangent or "Tan" of an angle can also be defined as: Tan (θ) = sin (θ) / cos (θ) but other than knowing there is more than one way to skin the stair-cat, you really don't want to go there while building a 38° stair so let's go back to work.

1. Find the Tangent of the Stair Angle that was Given

We find Tan 38° = 0.7813 using our calculator's Tan button, by looking up a TANGENTS TABLE or Tangents calculator on line, or we go back to high school and take our trigonometry class again to re-learn how to calculate the tangent.

2. Write the Tangent Formula Relating the Stair or Step Rise & Run

On my drawing I've marked X₁, the stair tread depth and I marked Y₁ as the vertical rise on one step.

To build this stair I'll pick a reasonable tread depth X₁ and will calculate the required step riser height Y₁

We write 0.7818 = Y₁ / X₁

We choose to make our individual stair tread depth 10" that's the horizontal "run" distance for one step. That's a nice run, right? And good planning for this example. Anybody can divide by ten.

3. Solve the Tangent Equation to Get the Stair or Step Rise

We can re-write the equation I just gave as

Y₁ = 10 x 0.7813 or

Y₁ = 7.8" - that's the riser height we'll use to build a 38° stair with a 10" tread depth.

Complete details about converting slope or angle to stair rise & run along with other neat framing and building tricks using triangles and geometry are found

at FRAMING TRIANGLES & CALCULATIONS.

How to Adjust the 38° Stairs to Fit the Building

You'll have to explain to your customer (or your wife if the 38° stairs were here idea) that you can either change the building space alotted to the stairway to fit the exact 38° angle stairs, or you can adjust the stair riser or tread depth slightly to fit the space presently available in the building.

Now to adjust our stairs to fit the exact building, whose stairway rise and run space might not fit the 38° stair we calculated, see

• CALCULATE TOTAL NUMBER OF STAIR STEPS & RISERS
• CALCULATE TOTAL STEPS
• CALCULATE STEP TREAD DEPTH

If you bought pre-built stairs see ADJUST FACTORY-BUILT STAIR to FIT the SITE

...

Continue reading at STAIR CONSTRUCTION, SPECIFIC TOTAL RISE & RUN - build stairs to a specified total rise AND total horizontal run distance, or select a topic from the closely-related articles below, or see the complete ARTICLE INDEX.

Or see these

Recommended Articles

• CALCULATE stair tread depth or riser height from stairway slope in degrees
• ROOF SLOPE CALCULATIONS explanation of the mathematics of calculating stair or roof slope, angle, pitch, rise, run, etc. + using trigonometric functions
• STAIR CONSTRUCTION, LOW ANGLE SLOPE
• STAIR CONSTRUCTION, SPECIFIC ANGLE or SLOPE
• STAIR CONSTRUCTION, SPECIFIC TOTAL RISE & RUN - using landscape ties as example
• STAIR DESIGNS for UNEVEN / SLOPED SURFACES
• STAIR DIMENSIONS, WIDTH, HEIGHT
  • STAIR ANGLE & SLOPE
  • TANGENT & SLOPE CALCULATIONS
  • STAIR DIMENSIONS, IDEAL
  • STAIR PLATFORMS & LANDINGS
  • STAIR RISE & RUN CALCULATIONS
  • STAIR RISER SPECIFICATIONS
  • STAIR TREAD DIMENSIONS
  • STAIR TREAD - HALTING WALK STUTTER STEP

Suggested citation for this web page

STAIR CONSTRUCTION, SPECIFIC ANGLE or SLOPE at InspectApedia.com - online encyclopedia of building & environmental inspection, testing, diagnosis, repair, & problem prevention advice.

Or see this

INDEX to RELATED ARTICLES: ARTICLE INDEX to STAIRS RAILINGS LANDINGS RAMPS

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Citations & References

In addition to any citations in the article above, a full list is available on request.

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