How this calculator works
When two circles overlap, their boundaries cross at two points and the shared region is a lens. Each crescent, or lune, is the part of one circle that lies outside the other. The calculator finds the lens area by adding two circular segments, one cut from each circle by the common chord, then subtracts that overlap from each full circle area. A true crescent requires partial overlap: if one circle is entirely inside the other, or the circles are apart or just touching, there is no crescent and the calculator asks for a different distance.
Formula
With centres distance d apart, the lens area is r₁² acos(x₁/r₁) - x₁√(r₁² - x₁²) + r₂² acos(x₂/r₂) - x₂√(r₂² - x₂²), where x₁ = (d² + r₁² - r₂²)/(2d) and x₂ = d - x₁. Crescent 1 = πr₁² - lens; crescent 2 = πr₂² - lens.
Worked example
Circles of radius 5 m and 3 m with centres 4 m apart have a lens area of about 18.2247 m². The larger crescent is then π × 25 - 18.2247 ≈ 60.3151 m² and the smaller is π × 9 - 18.2247 ≈ 10.0496 m².
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Frequently asked questions
Which circle gives which crescent?
Crescent 1 is the part of circle 1 outside circle 2, so it uses the first radius. Swapping the two radii swaps the two crescent areas but leaves the overlap unchanged.
What if one circle sits completely inside the other?
The smaller circle is not cut by the larger boundary, so no crescent is formed. The calculator reports an error instead of returning a negative or zero crescent.
Why must the distance be strictly between the radius difference and the sum?
At the difference the circles touch internally, and at the sum they touch externally. In both cases the crescent has zero width at a single point, which is a degenerate shape rather than an area.
Is a crescent the same as a lune?
In this calculator yes. A lune is the region bounded by two circular arcs, which is exactly the visible crescent left when one circle covers part of another.