Topic 2 – Perform Mathematical Calculations for Stairs

Unit Rise

The first most important consideration to make when calculating a set of stairs is the total rise. Every component of a set of stairs is dependent on the total rise and the amount and height of each unit of rise. The first step for calculating a set of stairs is finding how many units of rise there are in a set of stairs. This is typically done by taking the total rise and dividing it by proposed rise.

When aiming for a comfortable set of stairs, the proposed rise is typically going to be 180 mm (or 7”). When we divide the total rise by this number, we will round up or down to the nearest value to find the number of rises. For example, we will find the number of risers for a set of private stairs with a total rise of 2800 mm:

Since the value resulting from dividing the total rise by 180 mm is 15.56, we will round it up to 16. This means that we have 16 units of rise. From there, we can find the value of the unit of rise by dividing the total rise by the # of risers:

At this point, it is worth checking to see that the unit rise falls within code. Checking section 9.8.4.1 in the NBC, we find that private stairs have a minimum rise of 125 mm and a maximum rise of 200 mm. Our value is well within these parameters.

In some instances, we may be asked to minimize the amount of risers and treads. In order to accomplish this, our proposed rise will be the maximum allowable rise (180 mm public stairs, 200 mm private stairs). When using the maximum rise as proposed, the same process as above will be followed, with the exception that the number of risers must always be rounded up. For example, we will use a total rise of 3050 mm in a set of private stairs (200 mm maximum rise):

We will always round up or down to the nearest whole mm. At this point, we know that the unit rise will be within code requirements and the maximum allowable unit rise has been obtained.

***Find the unit rise for the following examples:

• Total rise 10’-6”, proposed rise 7 ¼”
• Total rise 2950 mm, maximum proposed rise public stairs

Unit Run (Total Run Given)

Unlike total rise, the total run of a set of stairs is not always predetermined. The total run may or may not be given. We will discuss both situations, both of which are dependent on first calculating the # of units of rise. Since the top floor acts as our last “tread”, the number of units of run is always one less than the number of units of rise.

If the total run is predetermined along with the total rise, finding the unit run is simply a matter of dividing the total run by the number of treads. We will use the following example and again first calculate the amount of and unit of rise to then find the number of treads.

• Total Rise = 2573 mm
• Total Run = 3415 mm
• Proposed Rise = 180 mm (private stairs)

Since we have 14 risers, we know that the number of units of run is one less, 13. We can take this number and directly divide the total run by its value:

It is always a good idea to double check your unit run and unit rise against the tables found in section 9.4.8.1 and 9.4.8.2 of the NBC. Looking into those sections, we find that these values are within code requirements.

Unit Run (Total Run Unknown)

When the total run is not given, we must use a specified formula in relation to the unit rise. These formulas are not without flaws as they may produce results that are outside of code requirements. When using them, the unit rise and run should always be double checked against code after calculating. The most common formula used is as follows:

As this formula is typically used to find unit run, it is rearranged as follows:

We can use the previous example above (Total Rise 2573, 14 Units of Rise @ 184 mm) to find the unit run:

If we check this value against code, we find that it does indeed fall between the minimum (255 mm) and maximum (355 mm) for private stairs. If required, we can also calculate the total run at this point by multiplying the number of treads by the unit run:

Calculating Stringer Length

To order material required for a set of stairs, we must approximate the length of material required for each stringer. There are two methods which can be used to find this value, both of which use Pythagorean’s theorem (a² + b² = c²).

The first is taking the total rise (a) and total run (b) to find the stringer length (c). We will use the example above with a total rise of 2473 mm and a total run of 3588 mm:

Another option uses the unit rise and unit run as a and b in Pythagorean’s theorem and multiplying the result by the number of units of rise (using # of units of run will yield a result too short for use). We will use a set of stairs with 15 units of rise at 187 mm and a unit run of 273 mm:

Review Questions

1. Fill in the following chart for a dwelling unit. Use 180 mm or 7” for a proposed rise.

2. Fill in the following chart to find the maximum allowable unit rise for a dwelling unit.

3. The number of treads is one (more/less) ______ than the number of risers.
4. How is the total number of risers determined for a stair?
5. How is the unit rise determined?
6. How is the number of units of run determined?
7. Total rise is 1600 mm with a proposed unit rise of 190 mm. (1600 mm ÷ 190 mm = 8.42 units of rise). If the remainder (0.42) is rounded up, does the stair become steeper than if the remainder is dropped?
8. How can the approximate length of a stringer be determined? (two methods)
9. Calculate the unit rise for the following stairs. Do not exceed the NBC maximum for a single dwelling unit.

10. Using the information in the following table, complete the calculations in the table on the next page:

Answer:

3. Less
4. Taking the total rise and dividing by proposed rise, rounding up or down

(Always round up if proposed rise = maximum allowable rise)

5. Taking the total rise and dividing it by the number of risers.
6. Subtracting 1 from the number of units of rise.
7. If the remainder is rounded up, the stairs become LESS steep.
8. 1. Calculate the diagonal of the unit rise and unit run and multiply by the number of risers
   2. Calculate the diagonal of the total run and total rise
9. 199. 199.17 mm
   200. 195 mm
   201. 178.57 mm