What is the formula for stairs

Calculate stairs: an overview of the most important formulas

If you want to calculate and plan a staircase, you are forced to take a few parameters into account: step height, step width, gradient ratio and much more. We show you the most important formulas for calculating stairs.

Definition: Terms for calculating stairs

  • Entry and exit: first and last step or tread area
  • Appearance: step depth
  • Incline: clear height between the steps
  • Increment: Stride length (from heel edge to heel edge)
  • Usable running width: distance between wall and handrail

DIN 18065 building stairs: regulation for the construction of stairs

DIN 18065 defines the rules for the incline, tread width and tread of stairs. However, the set of rules is not part of the building regulations - in other words, extra requests on the part of the client are quite conceivable, especially for single-family houses.

The regulation on stair construction provides for a distinction between stairs required under building law and stairs not required under building law. Basically, all stairs that serve to reach a residential floor or as an escape route are necessary. For example, there is no need for stairs to access the attic with a pure function as a storage facility.

1. Necessary stairs

  • Usable running width: at least 80 centimeters
  • Incline: at least 14, maximum 20 centimeters
  • Appearance: at least 23, maximum 37 centimeters

2. Unnecessary stairs

  • Usable running width: at least 80 centimeters
  • Incline: at least 14, maximum 20 centimeters
  • Appearance: at least 21, maximum 37 centimeters

Formulas for calculating stairs

Before you start calculating the stairs, you will most likely only know the height to be overcome. Based on this value, you can determine the number of steps, the height of the step, the width of the step, the length of the run and the gradient of the stairs.

Step 1: Calculate the number of steps

You can find out the number of steps on the new staircase by dividing the height of the storey by the height of the slope (guide value: 18 centimeters). You round the result up or down accordingly.

Floor height / incline height = number of steps

Good to know: In stair construction it is generally said that a staircase should not have more than 18 steps (conventional storey height up to approx. 3 meters). In the case of a higher number of stairs, a landing is traditionally installed in the middle of the stairs.

Step width + number of steps x 65 centimeters = platform length

Step 2: calculate the height of the slope

After you have determined the number of steps, you can determine the height of the stairs in order to always achieve the same distances - goodbye to stumbling blocks!

Floor height / number of treads (rounded) = gradient height

Step 3: Calculate the tread width of the stairs

With the step width of the stairs you determine the depth of the step: How much space does the foot need on a step? You can use 29 centimeters as a guideline. To calculate the step width, you need the step size. Please note that the step depth plus double the step height results in an amount between 61 to 65 centimeters. Because 63 centimeters are the average step of an adult - suitable for stairs with an angle of inclination between 30 and 37 degrees. Attention: With a higher incline the steps become too narrow, with a smaller incline too wide! In order to determine the step width, subtract the pitch height twice from the increment.

Incremental measure (approx. 63 centimeters) - 2 x incline height = step width

Step 4: calculate the run length

To calculate the running length, multiply the tread width (guideline: 27 centimeters) by the number of steps.

Entrance width x number of appearances = run length

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Step 5: Calculate the slope of the stairs

To consider the slope of the new staircase, divide the floor height by the planned length of the staircase. If the result is higher than 1, the stairs are too steep. A good alternative would be a spiral staircase, for example. If the result is less than 0.45, you can increase the slope.

Floor height / running length = stair incline

Step 6: Security rule and convenience rule (DIN 18065)

Finally, use the safety rule to check whether the step width is not too short - otherwise a dangerous fall is inevitable! Attention: The sum of tread and incline should ideally show 46 centimeters.

Security rule:
Incline height + step width = approximately 46 centimeters

You can then use the convenience rule to control the comfort of your stairs. The difference between step and slope should be 12 centimeters - suitable for stairs with a 30 degree incline.

Convenience rule:
Step width - slope height = approximately 12 centimeters

Reading tip: We'll tell you how to carefully plan and install stairs.