Staircase calculations

Calculation of the gradient of a staircase, based on headroom and dimensions of the staircase well, as used in the Staircase tool (see below).

Terminology

The following table lists the terms used to describe the staircase, in english and dutch.

p	Stair well length	Trapgatlengte
q	Headroom + floor thickness	Doorloophoogte + vloerdikte
t	Tread depth	Trede breedte
h	Rise unit	Optrede
α	Gradient	Stijgingshoek
n	Nosing	Neus

Nosing

At first thought one might think the angle of the staircase is simply determined by the ratio q : p. But since generally the tread depth is larger than the run unit, the treads overlap each other. This overlap is called nosing. The following figure depicts this situation, where n denotes the depth of the nosing.

So we have

where the last term

equals the run unit.

Gradient

The next figure shows the situation with nosing accounted for:

The gradient depends on the nosing, while we saw above that the nosing in its turn depends on the gradient. We might suspect complications because of this mutal dependency, but it turns out to be quite simple. From the figure we read

so using the earlier formula for n

which simplifies to

We can also illustrate this formula with a slightly different figure, in which we position the first step from the top:

Tool

This concludes the explanation of the formulas used in the code of the on line staircase calculator.