L-Shaped Winder Stair (Quarter-Turn)
About L-Shaped Winder Stair Calculation
This L-Shaped Winder Stair and Quarter-Turn Staircase Calculator performs a geometric calculation of an L-shaped staircase with a 90° quarter-turn and winder steps. From the specified overall dimensions and number of steps, it determines the rise height, straight-tread depth, flight angle, comfort value, winder-section dimensions, stringer dimensions, and riser sizes.
The calculation is intended for selecting staircase proportions and preparing dimensioned drawings. The straight steps of the upper and lower flights are calculated with the same effective tread depth, while the winder steps are generated by geometrically dividing the 90° turn. The resulting geometry can be used when planning both wooden and metal staircases.
Guidelines and Recommendations
Automatic selection of the number of steps
Search principle. When automatic selection is enabled, the calculator checks all permitted combinations of upper straight steps, winder steps and lower straight steps. Only geometrically valid variants are considered.
Evaluation criteria. For each variant, the calculator determines the step height h, calculated step depth a, flight angle α and comfort value 2h + a. The target ranges are 150–200 mm, 270–320 mm, 30–40° and 600–660 mm respectively.
Selection of the best variant. Priority is first given to combinations with the smallest deviation outside the target ranges, then to variants with fewer parameters outside those ranges and with values closer to the centers of the target ranges. If several variants remain equivalent, the calculator prefers a number of winder steps closer to 3 and a more balanced distribution of the available step depth between the upper and lower flights.
If the given staircase dimensions do not allow all target ranges to be satisfied simultaneously, the algorithm still selects the closest geometrically valid variant. In this case, one or more result values may remain outside the recommended ranges and are highlighted by the calculator.
Manual editing of the number of upper, winder or lower steps disables automatic selection. If automatic selection remains enabled, changing the staircase dimensions causes the number of steps to be selected again automatically.
Number of rises and step height
Total number of rises. The floor-to-floor height is divided into equal vertical increments. The calculation includes the lower straight steps, winder steps, and upper straight steps. The position of the final step relative to the upper-floor level determines whether an additional rise is required.
n = nlower + nwinder + nupper + k
Here, nlower, nwinder, and nupper are the numbers of the corresponding steps, while k = 1 if the upper step is below the second-floor level and k = 0 if it is level with the upper floor.
Step height. All rises, including those through the winder section, are assumed to have the same height. For a total staircase height H in millimetres, the height of one rise is:
h = H / n
Therefore, changing the number of steps immediately changes the rise height and subsequently affects the staircase angle, inclined step spacing, and comfort value. The calculator uses 150–200 mm as a practical guideline for a comfortable rise height.
Depth of straight steps
Available length of the upper flight. The flight width M, occupied by the turning zone, is subtracted from the overall staircase length L. The remaining distance is distributed among the upper straight steps:
aupper = (L - M) / nupper
Available length of the lower flight. In the same way, the turning-zone width M is subtracted from the overall staircase width B, and the remaining distance is divided by the number of lower straight steps:
alower = (B - M) / nlower
Common calculated tread depth. The calculator uses the same going for both straight flights. Therefore, the smaller of the two available values is selected:
a = min(aupper, alower)
This ensures that straight steps with the same tread depth fit within both the available staircase length and width. If one of the straight flights is absent, the flight width M is used as the limiting geometric value for that direction.
For the calculated going, the calculator uses 270–320 mm as a practical guideline. This value is the horizontal distance between successive rises and does not include the tread nosing.
Full tread depth
Tread nosing. The visible tread can be deeper than the calculated going because the front edge may project beyond the riser. The specified nosing s is therefore added to the effective tread depth a.
Riser thickness. If risers are included, their thickness tr is also included in the full geometric depth of the tread component. The resulting tread depth is:
afull = a + s + tr
If risers are not used, tr = 0, so the full tread depth is a + s. This is why the physical tread component can be deeper than the calculated going.
Staircase angle and inclined step spacing
Flight angle. After the rise height h and horizontal going a are known, the inclination of the straight flights is calculated from a right triangle:
α = arctan(h / a)
The same angle is applied to the lower and upper straight flights because they use the same rise height and the same calculated tread depth. The calculator uses 30–40° as a practical guideline.
Inclined distance between steps. The length of one step interval along the stringer is calculated using the Pythagorean theorem:
l = √(h2 + a2)
This value is then used when calculating the lengths of the straight inclined sections.
Comfort value 2h + a
Relationship between rise and going. An additional check is performed using the commonly used staircase proportion formula:
P = 2h + a
The formula relates two vertical rises to one horizontal going and provides a combined assessment of step proportions. The calculator uses 600–660 mm as a comfortable reference range, with approximately 630 mm as the recommended central value.
The 2h + a check supplements rather than replaces the separate assessment of rise height, tread depth, and flight angle. For example, an acceptable result from the comfort formula does not by itself mean that an excessively high or narrow step has suitable proportions.
Winder turn geometry
Angular division. The total turn is always 90°. It is divided into equal angular parts according to the number of winder steps:
β = 90° / nwinder
With three winder steps, the basic angular increment is 30°; with four, it is 22.5°; with five, it is 18°. The calculator allows from 2 to 8 winder steps; three steps are commonly used for a 90° quarter-turn staircase.
Winder-step geometry. The outlines of the individual winder steps are generated sequentially inside the turning zone using the flight width and equal angular division. Each step shape is obtained from the intersection of the corresponding radial directions with the boundaries of the turning area, so changing the number of winder steps changes both their quantity and the geometry of each individual tread.
Turning-zone limit. The flight width M is used as the basic dimension. If twice the flight width exceeds the overall staircase width B, the corresponding turning-zone dimension in the drawing is limited to half of the total width:
Mturn = B / 2 when 2M > B
This condition keeps the turning geometry within the specified overall width. The tread nosing is also taken into account when constructing the outer boundary of the winder section.
Height at the start of the winder section. The vertical position of the turn is determined by the number of lower straight rises:
Hwinder = nlower · h
This value gives the height at which the base of the winder-step group begins above the lower floor level.
Stringer calculation
Lower straight section. For the lower stringer, the upper-edge length is first determined from the number of lower steps, rise height, tread thickness t, and staircase angle α:
Llower,upper = (nlower · h - t) / sin(α)
Opposite edge of the lower stringer. Because the stringer has a specified width w, its opposite edge receives a geometric correction based on the inclination angle:
Llower,lower = Llower,upper - w · tan(α) - w / tan(α)
The width of the stringer therefore directly affects the difference between the two longitudinal dimensions of the lower component.
Upper straight section. For the upper section, the inclined spacing of one step l is multiplied by the number of upper steps:
Lupper = nupper · l
In the current geometric model, both edges of the upper straight stringer are taken as the same calculated length Lupper. This is an assumption used by the calculator for this part of the drawing.
Riser calculation
Riser height. A riser fits between two adjacent treads, so the tread thickness is subtracted from the full rise height:
hr = h - t
For example, with a rise of 180 mm and a tread thickness of 40 mm, the calculated riser height is 140 mm.
Number of risers. The number of risers is taken as equal to the total number of rises n. The length of each riser is taken as the specified flight width M.
Calculation limits
Geometric compatibility. For the staircase geometry to be constructed, the overall length and width must not be smaller than the flight width. The tread thickness must be smaller than the calculated rise height, and the lower-stringer dimensions must remain positive after the angular correction is applied.
Number of steps. The calculation supports up to 13 straight steps separately for the upper and lower flights and from 2 to 8 winder steps. All intermediate geometric values are recalculated whenever the number of steps changes.
European references
EN 17210:2021 “Accessibility and usability of the built environment — Functional requirements”. This European standard establishes general principles for accessibility and usability of elements of the built environment, including circulation routes and stairways. Final staircase dimensions should also be checked against the requirements applicable to the specific building type and country.
CEN/TR 17621:2021 “Accessibility and usability of the built environment — Technical performance criteria and specifications”. This document supplements the functional approach of EN 17210 with technical criteria used when assessing accessibility and usability of the built environment.
The ranges 150–200 mm, 270–320 mm, 30–40°, and 600–660 mm used by the calculator to assess proportions are practical design guidelines. Specific permissible staircase dimensions in European countries may additionally be governed by national building requirements.
FAQs
Why is the tread depth the same for the upper and lower flights?
An L-shaped staircase must fit within the available dimensions in both directions. The calculator separately determines the possible tread depth for each direction and then selects the smaller value. This gives both straight flights the same going while keeping the quarter-turn staircase within the specified overall size.
Why does changing the number of steps affect several results at once?
The number of steps determines the total number of vertical rises, so the rise height h changes first. The calculator then recalculates the tread depth, staircase angle, inclined step spacing, 2h + a comfort value, and stringer dimensions. The staircase geometry is therefore calculated as one connected system rather than as a set of independent dimensions.
How does the number of winder steps affect the quarter-turn staircase?
The total turn remains 90°, but this angle is divided among all winder steps. Increasing their number reduces the angular sector assigned to each tread and changes the shape of the complete turning section. Three winder steps are a common arrangement for an L-shaped staircase with a 90° quarter-turn.
Why is the full tread depth greater than the calculated going?
The calculated going a is the horizontal distance between successive rises. The physical tread also includes the front nosing and, when risers are used, the riser thickness. Therefore, the tread component is calculated as afull = a + s + tr.
What changes when the top step is level with the second floor?
The number of vertical rises used to divide the total staircase height changes. If the top step is below the second-floor level, one additional rise is included; if it is level with the upper floor, no extra rise is required. This changes the rise height, staircase angle, and the related calculated dimensions.