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Spiral Stair Tread and Helix Geometry

Estimate equal stair rises, tread angles, radial lengths and ideal helical stringer/handrail lengths.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

In initial displayed units, Inside diameter 40, Outside diameter 200, Total rise 3, Maximum riser rise 20, Angle of rotation 360 gives Number of steps 15, Number of treads 14, Actual riser rise 20, Tread length 80.

Rises = ceil(total rise/maximum rise); treads = rises−1; tread angle = rotation/treads; helix length = treads×√(arc²+rise²).

Worked example

Enter these known values and leave the other values blank.

Inside diameter
40 cm
Outside diameter
200 cm
Total rise
3 m
Maximum riser rise
20 cm
Angle of rotation
360 deg
Number of steps
15 steps
Actual riser rise
20 cm
Inner arc
8.976 cm
Outer arc
44.88 cm
Tread depth check
24.31 cm
Tread angle
25.714 deg
Tread length
80 cm
Number of treads
14 treads
Handrail length
6.879 m
Inner stringer length
3.069 m
Inner stair angle
65.83 deg
Outer stair angle
24.02 deg
Headroom clearance
2.8 m

Assumptions and limitations

  • The model uses N rises and N−1 treads; the upper landing is excluded. The 12-inch depth line and initial headroom limit are retained planning reference conventions, not jurisdiction-specific requirements. Individual tread angle must stay below 90° for the tangent-depth representation. Exact whole-count operations accept finite quantities with magnitude below 10^50.
  • Use compatible measurement units. Clear an unknown before solving for it, and lock independent known values when comparing changes. Currency choices label amounts without exchange-rate conversion.

Common questions

Can I solve backward from the result?

Clear the unknown and enter independent known values. A rounded whole count cannot uniquely recover a continuous dimension; keep the original measured dimensions for procurement planning.

References

Use exact decimal ceilings for whole quantities because the captured mathjs ceiling can undercount immediately above an integer boundary. Enforce the stated physical geometry and whole-count domains.

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