Free · Geometry and function relationships

Circular Segment Area, Chord and Height

A circular segment is bounded by a chord and its arc. The central angle in radians gives sector area minus the triangle area; minor and major segments use different angle branches.

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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.

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Formula and method

For radius = 5, angle = 60, chord_length = 5, segment_area = 2.264651842651988742381141129252893057429006327748061995380142182.

A=r²(θ−sin θ)/2; chord=2r sin(θ/2)

Worked example

Enter these known values and leave the other values blank.

Radius (r)
5 cm
Central angle (α)
60 deg
Height (h)
0.67 cm
Arc length (L)
5.24 cm
Chord length (c)
5 cm
Segment area (A)
2.265 cm²

Assumptions and limitations

  • A circular segment is bounded by a chord and its arc. The central angle in radians gives sector area minus the triangle area; minor and major segments use different angle branches.
  • Clear the unknown field before solving. Locks preserve known quantities; an inverse may have multiple solutions and this model selects its principal branch.
  • Numerical rounding does not describe measurement uncertainty. Extremely large arguments or near-singular inverses can exceed the available precision; use moderate values for interpretable finite results.

Common questions

How do I enter a consistent set of values?

A circular segment is bounded by a chord and its arc. The central angle in radians gives sector area minus the triangle area; minor and major segments use different angle branches.

Can I calculate a missing input?

Clear that field and provide enough independent known quantities. Output-only fields and nonunique inverse branches may need a different mode.

References

A circular segment requires a positive radius and positive central angle, with nonnegative lengths and area. The captured source only checks upper angle and height bounds.

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