Free · Circle calculators ⭕

Circular Sector Area, Arc and Chord Relationships

Calculate circular sector area, arc and chord relationships with retained unit controls, positive geometric roots and explicit inverse-branch assumptions.

  • Formula & worked example
  • Private in your browser
  • No signup
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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

Worked inputs Central angle (⍺) = 90, Radius (r) = 2 give Diameter (2r) = 4, Sector area (A) = 3.1415927, Arc length (L) = 3.1415927, Chord length (c) = 2.8284271. Values use the initial displayed units.

L = rθ; A = r²θ/2; chord = 2r sin(θ/2)

Worked example

Enter these known values and leave the other values blank.

Central angle (⍺)
90 deg
Radius (r)
2 cm
Diameter (2r)
4 cm
Sector area (A)
3.14 cm²
Arc length (L)
3.14 cm
Chord length (c)
2.83 cm

Assumptions and limitations

  • The entered angle may describe a minor or major sector strictly below one full turn. A chord and radius alone do not distinguish both sectors; the captured chord inverse returns the minor angle.
  • Lengths and areas are converted before calculation. Clear an unknown measurement instead of supplying conflicting dimensions.
  • Arithmetic uses 64 significant digits and full-precision π. Display rounding and measurement uncertainty are separate from the underlying arithmetic.

Common questions

Which measurements and inverse branch does this model use?

The entered angle may describe a minor or major sector strictly below one full turn. A chord and radius alone do not distinguish both sectors; the captured chord inverse returns the minor angle.

Can a chord identify a major sector by itself?

No. A chord and radius generally admit a minor and a major angle. The retained chord inverse returns the minor angle; enter angle or arc length to identify the major sector.

References

The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor. The optional legacy US survey foot uses exactly 1200/3937 meters, replacing the rounded reference factor. Ordinary international feet remain exactly 0.3048 meters.

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