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Circular Arc Chord & Sagitta Calculator

Find the minor-arc sagitta from a circular radius and straight chord width, with an illustrated measurement guide.

  • 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

For a circle of radius 5 m and a straight chord width of 6 m, the minor-arc sagitta is 1 m. It is the perpendicular gap from the chord midpoint to the arc.

s = R − √(R² − (D/2)²), where D is the straight chord or aperture width

Worked example

Enter these known values and leave the other values blank.

Circle radius R
5 m
Straight chord / aperture width D
6 m
Minor-arc sagitta s
1 m

Assumptions and limitations

  • Models a circular minor arc. The chord is a straight width, not arc length or the full circle diameter unless it spans a semicircle.
  • Radius, chord and sagitta must be positive; the chord cannot exceed twice the radius. The minor-arc branch has sagitta no greater than the radius.
  • This geometric depth does not predict material deformation, structural deflection or load capacity.

Common questions

What happens when the chord equals the circle diameter?

The arc is a semicircle. Its sagitta equals the radius: a 10 m chord on a 5 m radius circle has a 5 m sagitta.

Can I solve for radius from a measured chord and sagitta?

Yes. Leave the radius blank and enter the chord plus a positive sagitta on the minor-arc branch. A 6 m chord and 1 m sagitta give a 5 m radius.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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