What is the height formula

Calculate stairs: Overview of the most important formulas and terms

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Incline ratio, run length or step - you should be familiar with these terms when dealing with the calculation of stairs. And without a proper staircase calculation, the steps will be of different sizes and the gradient ratio will not be correct. If this happens in the planning, builders misjudge the space around the stairs. However, if incorrect planning is implemented, the risk of accidents increases considerably.

The rough and detailed planning of a staircase is usually done by architects and stair builders or carpenters. But building owners who assign individual trades for their new building and also take care of the staircase planning, or owners who also install a new staircase during renovation, should know certain basic concepts of staircase calculation.

Too steep, too narrow or even too wide? With our stair calculator you can calculate the incline, number of steps or tread width of your stairs.

Building regulations and DIN 18065

The right terms are not only important so that everyone involved knows what is being talked about. But also because they are used in the regulations that define how a staircase must be designed. These include primarily the building regulations of the state building regulations, as well as the requirements of DIN 18065. Both specify the dimensions for the running width, incline and tread of a staircase or determine the height of the banister. Although DIN 18065 does not have the character of a law, it describes the generally recognized rules of technology that a stair builder must generally apply.

Incline, step and tread width

The three most important terms are the incline, the step width and the tread width.

  • Under the Pitch a staircase means the height of the individual steps. A step height between 16 and 18 centimeters is perceived as particularly comfortable and safe.
  • Stair treads are called appearance in technical jargon. The Tread width refers to the tread, seen from the side. If, on the other hand, you look at the step from the perspective of a person who is currently using the stairs, step depth would be the correct word. 29 centimeters are considered a good guide.
  • The term Running width again actually refers to the perspective of the stair user and means the width of a stair step, i.e. the distance between the left and right edge of the stair. In single-family houses, the running width is usually between 80 and 100 centimeters.

There are also a few other terms that are particularly important for implementation planning. They are less important when planning how much space a staircase will take up in the floor plan.

  • The Increment is in the end the height of two steps, i.e. twice the incline height. It is the height that you can climb with one leg when climbing stairs.
  • The Barrel length results from the sum of all tread widths. The Floor height is set by the rest of the house planning, but corresponds to the sum of all step or slope heights.
  • The Pitch angle results from the length of the run and the floor height, or the sum of all incline heights.
  • The headroom or the clear passage height denotes the distance between a step and the floor. DIN 18065 stipulates a minimum height of 200 centimeters - which, however, can be too low depending on the size of the residents. Usually a headroom of at least 210 centimeters is provided.
  • The Stair opening results from the run length and the clear passage height. It describes the area that is lost on the upper floor through the stairs.


When planning the handrail or the banister, the regulations of the state building regulations as well as DIN 18065 must be complied with. For a fall height of less than twelve meters, a railing with a height of at least 90 centimeters must be used, for more than twelve meters it is even 110 centimeters. This means that modern stairs, as you can see them in some magazines or brochures, cannot be implemented at all unless a handrail is planned.

If there are children in the house, the openings in the railing, for example between the individual struts, should not be wider than twelve centimeters. A glass plate in front of a cross-braced railing prevents children from using it as a ladder.

Calculate stairs: the most important formulas

Various formulas help with the stair calculation, including safety, comfort and step-by-step measures. A step height of 17 centimeters in relation to a step width of 29 centimeters follows all three rules.

Calculate run length

How long a straight staircase will be can be easily calculated: the step width multiplied by the number of steps. However, there should be at least one meter of space for entry and exit on each staircase at the top and bottom.

Run length = width of appearance x number of appearances

Calculate number of stages

To calculate the number of steps on the new staircase, divide the floor height by the required slope height, i.e. the individual stair step height. This is rounded up or down.

Number of steps = floor height / incline height

Calculate the height of the slope

Once you have determined the number of steps, you can now calculate the exact height of the stairs. A height between 16 and 18 centimeters is considered comfortable.

Exact gradient height = floor height / number of steps

Calculate the tread width of the stairs

The tread width describes the depth of a step, i.e. how much space the foot has on the step. We recommend around 29 centimeters here. For the formula, the increment is required, which is 63 centimeters on average.

Step width = 63 centimeters - (2 x incline height)

Calculate the slope of the stairs

If you divide the height of the storey by the planned length of the stairs, you get a value that indicates the slope. If this is greater than 1, the stairs are too steep. In this case, there are other models of stairs, such as spiral staircases, that are more suitable for the space available. If the value is below 0.45, the stairs can be a little steeper so as not to waste unnecessary space.

Incline = floor height / run length

General rules for building stairs

In addition to the formulas for calculating the individual stair elements, there are general rules that are used for stair planning.

Step-by-step rule

It is recommended that the depth of a stair step plus double the step height add up to 61 to 65 centimeters. 63 centimeters correspond to the step size of an average-sized person. This formula is particularly suitable for stairs with an angle of inclination between 30 and 37 degrees. If the slope is higher, the steps are too narrow, if it is less than 30 degrees, it is too wide.

2 x rise height + step width = about 63 centimeters

Security rule

The safety formula is used to check whether the step width is not too small and thus encouraging falls. The sum of the tread and the incline should be 46 centimeters.

Incline height + step width = 46 centimeters

Convenience rule

While the safety formula is particularly taken into account in public buildings, home builders tend to pay attention to easy ascent and descent. The difference between the tread and the slope should be 12. This rule can also be used for stairs with an incline of about 30 degrees.

Step width - rise height = 12

If you want to consider safety, comfort and the step size formula equally, you are exactly right with a staircase with a 17 centimeter rise and 29 centimeter step depth: These are the only dimensions that actually correspond to all three formulas.

Use space below and above the stairs

If you want to save on house building and build as cheaply as possible, you should save space. And that means that even the smallest rooms have to be used. With the stairs, the space below and above the stairs is ideal. By placing a cloakroom or a storage room under the stairs, for example, or by using the space above the stairs through the smallest possible stair opening.

The opening of the stairs can be determined by a two-step calculation:

  • Floor height - pitch height * X> passage height
    This means that the planned floor height, minus the incline of a certain number of steps (X), must be greater than the desired passage height.
  • Run length - X * step width = stair opening
    The determined number of steps (X) must be multiplied by the tread width and subtracted from the length of the stairs. The result is the length of the stair opening, the width of the stair opening corresponds to the width of the stairs.

The space under the stairs depends on the distance and the angle of incline of the stairs, but also on how the stairs are built in detail. For example, if the staircase is closed, the space can be used better than if it is an open staircase, in which the space between the individual steps is open.

Matthias Dittmann

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