Free · Geometry and function relationships

Intersecting Circles and Crescent Areas

These formulas apply to two circles whose boundaries cross twice. Each crescent equals its disk area minus their common overlap; concentric and separate disks need another geometric case.

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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_1 = 2, radius_2 = 2, circle_center_distance = 2, overlap_area = 4.913478794435027382178823005733596290640174557379520933154904955, crescent_area_1 = 7.652891819924145571671750527384415246148503040120902350744873415.

crescent₁=πr₁²−overlap

Worked example

Enter these known values and leave the other values blank.

Radius of circle 1 (r1)
2 cm
Radius of circle 2 (r2)
2 cm
Circle center distance (d)
2 cm
Area of lune 1
7.65 cm²
Area of lune 2
7.65 cm²
Overlap area
4.91 cm²

Assumptions and limitations

  • These formulas apply to two circles whose boundaries cross twice. Each crescent equals its disk area minus their common overlap; concentric and separate disks need another geometric case.
  • 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?

These formulas apply to two circles whose boundaries cross twice. Each crescent equals its disk area minus their common overlap; concentric and separate disks need another geometric case.

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

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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