Straight Single-Flight Stair

About Straight Single-Flight Stair Calculation

The straight single-flight staircase calculator determines the geometry of a straight stair: rise height, horizontal going, full tread depth, stair angle, comfort index, stringer or stair beam dimensions, and riser dimensions. The calculation assumes equal rise heights and equal horizontal goings along the entire stair flight and can be used when designing a straight single-flight stair for a private house.

Guidelines and recommendations

Automatic selection of the number of steps

When automatic selection is enabled, the calculator checks every integer number of steps from 1 to 30. For each variant, it determines the number of rises N according to the position of the top step, then calculates h = H / N, a = L / n, the stair angle α, and the comfort value 2h + a.

Variants with invalid geometry are discarded. The remaining variants are compared with the recommended ranges used by the calculator:

• rise height h - 150-200 mm
• horizontal step a - 270-320 mm
• stair angle α - 30-40°
• comfort value 2h + a - 600-660 mm

Priority is given to a variant that keeps all values within these ranges. If this is impossible for the entered dimensions, the calculator selects the combination with the smallest overall deviation. Values outside the recommended ranges are still highlighted in red, so automatic selection does not mean that the entered staircase dimensions are optimal.

If the number of steps is changed manually, automatic selection is disabled. It can be enabled again with the corresponding option.

Basic geometry and number of rises

Main dimensions. In the formulas, H is the total stair height in mm, L is the horizontal stair length in mm, n is the specified number of treads, and B is the stair flight width in mm. The tread thickness is denoted by t, the tread nosing projection by o, and the stringer or stair beam width by k. All linear dimensions are calculated in millimetres.

Number of rises. This depends on the position of the top tread. If the top tread is level with the second floor, the number of rises equals the number of treads:

N = n

If the top tread is below the second-floor level, one additional rise remains between the top tread and the floor:

N = n + 1

The total stair height is divided by the number of rises N, rather than simply by the number of treads.

Rise height and horizontal going

Rise height. The total height is divided equally between all rises:

h = H / N

Changing the number of treads therefore changes the height of each rise automatically. All rises are assumed to have the same height.

Calculated going. The horizontal length of the staircase is divided by the specified number of treads:

a = L / n

The value a is the horizontal distance between consecutive rises and is used to calculate the stair angle and the comfort index.

Stair angle

Stair flight angle. The inclination is calculated from the ratio between the rise height and the horizontal going:

α = arctan(h / a)

The greater h is relative to a, the steeper the staircase becomes. The calculator uses 30-40° as a practical comfort range. This range is used only to evaluate the result and does not alter the calculated geometry.

Full tread depth

Tread size. The full construction depth differs from the horizontal going because the tread nosing projection is included. If risers are used, their thickness is also included. The calculation uses:

reff = r with risers, reff = 0 without risers

b = a + o + reff

Here, b is the full tread depth in mm, o is the nosing projection, and r is the riser thickness. The full tread depth can therefore be greater than the calculated horizontal going a.

Comfort index 2h + a

Step rule. The relationship between rise height and going is evaluated using the expression commonly known as Blondel's formula:

K = 2h + a

The calculator treats 600-660 mm as a comfortable range, with 630 mm used as the recommended target. The formula uses the calculated horizontal going a, not the full tread depth including the nosing. Increasing only the nosing projection therefore does not improve the 2h + a value.

Combined evaluation. The result is also compared with practical ranges of 150-200 mm for rise height, 270-320 mm for horizontal going, and 30-40° for stair angle. These values are practical design guidelines used by the calculator. They do not restrict the calculation and do not replace national building requirements applicable to a particular building.

Risers

Riser board height. When risers are included, the tread thickness is subtracted from the full rise height:

hr = h - t

The resulting value hr is the height of the riser board between adjacent treads. The number of risers is taken as the number of rises N, and the length of each riser equals the stair flight width B.

Stringer or stair beam calculation

Inclined step distance. To construct the supporting member geometrically, the distance between adjacent steps along the stair slope is calculated:

s = √(h2 + a2)

This value combines the vertical rise and horizontal going and is used when drawing the staircase geometry.

Upper-side length. First, the inclined length of the upper side of the stringer or stair beam is calculated with allowance for tread thickness:

ltop = (h × n - t) / sin α

The expression h × n - t defines the calculated vertical height of the relevant section, while division by sin α converts this height into an inclined length.

Lower-side length. The width of the supporting member k and the geometry of its top and bottom cuts are then taken into account:

lbot = ltop - k × tan α - k / tan α

The upper and lower edges of the same blank therefore have different calculated lengths. The stringer width changes this difference but does not affect the rise height, horizontal going, or stair flight angle.

Calculation accuracy

Rounding. The formulas use the original input values without preliminary rounding of intermediate results. Linear dimensions are displayed to the nearest 1 mm, while the stair angle is displayed to 0.1°.

Stair direction. Selecting a right-hand or left-hand staircase mirrors the drawings but does not change the numerical calculation results.

Related European standards

EN 17210:2021 Accessibility and usability of the built environment. Functional requirements. This standard covers requirements for accessible and safe use of the built environment, including vertical circulation, steps, and stairways. Its requirements are functional in nature, so specific geometric dimensions may be further defined by the national regulations of the country where the staircase is built.

CEN/TR 17621:2021 Accessibility and usability of the built environment. Technical performance criteria and specifications. This technical report complements EN 17210 and provides technical criteria for implementing its requirements, including criteria related to steps and stairways. Final dimensions should therefore be checked against the building type and the applicable national requirements.

FAQs

Why can the number of rises be one greater than the number of treads?

If the top tread is below the second-floor level, another rise remains between that tread and the finished floor. The total height is therefore divided into n + 1 rises. If the top tread is level with the second floor, the number of rises equals n.

What is the difference between the calculated going and the full tread depth?

The calculated going a depends only on the horizontal stair length and the number of treads. The full tread depth b additionally includes the tread nosing and, when risers are present, the riser thickness. This is why the two values shown by the straight staircase calculator can differ.

Why does the comfort formula use 2h + a instead of the full tread depth?

The step formula evaluates the geometry of movement between consecutive rises, so it uses the horizontal going a. The nosing increases the physical depth of the tread but does not increase the horizontal spacing between consecutive rises. It is therefore not included in the 2h + a comfort index.

How should I choose the number of treads for a more comfortable staircase?

For a fixed total height, increasing the number of rises reduces h, while changing the number of treads also changes the horizontal going according to a = L / n. A practical approach is to compare several neighbouring tread counts and choose the option where the rise height, going, stair angle, and 2h + a value are all closest to their recommended ranges.

Why are the upper and lower stringer lengths different?

A stringer or stair beam has a finite width, so after the end cuts are formed its upper and lower edges begin and end at different points. The difference is calculated from the member width k and stair angle α. A wider supporting member or a different stair angle changes the difference between these two lengths.