Crescent area calculator
Find both lunes and the lens shared by two circles that cross at two points.
Formula
x=(d²+r₁²−r₂²)/(2d); y=d−x; h=√(r₁²−x²); I=r₁² arccos(x/r₁)+r₂² arccos(y/r₂)−dh; Lᵢ=πrᵢ²−I.
Radius r₁: 4 cm · Radius r₂: 6 cm · Center distance d: 5 cm · Lune 1 area ≈ (pi×(4 cm × 0.01 m/cm)^2-(0.002930279402))×10000 · Lune 2 area ≈ (pi×(6 cm × 0.01 m/cm)^2-(0.002930279402))×10000 · Overlap area ≈ (0.002930279402)×10000
How to use
- Enter both radii.
- Enter center-to-center distance d.
- Read each circle's uncovered lune and the shared lens.
Examples
- For r₁=r₂=d=1 cm, each lune is ≈1.913223 cm²; overlap is ≈1.22837 cm².
Frequently asked questions
Which circles are allowed?
Use |r₁−r₂| < d < r₁+r₂. Tangency and values very near it are excluded.
Which result is lune 1?
The part of circle 1 outside circle 2; lune 2 is the opposite part.
Is every lune a crescent?
In the strict sense, a crescent excludes its circle's center. This tool also computes other lunes.
Sources
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