Enter the radius of the main circle (r₁), the radius of the covering circle (r₂), and the distance between centers (d) to calculate the crescent (lune) area. You'll also get the overlapping area and the second lune area — all based on the precise two-circle intersection formula.
Crescent / Lune 1 Area
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Overlap Area
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Lune 2 Area
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Main Circle Area (πr₁²)
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Covering Circle Area (πr₂²)
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Shape Classification
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A crescent is a specific type of lune — a concave-convex region formed by two circular arcs. It is defined as a lune that does not contain the center of the original (main) circle. This occurs when the distance between the two circle centers is less than the radius of the covering circle.
A lens is a convex-convex region where both circular arcs curve outward. A lune is a concave-convex region where one arc curves outward and the other curves inward. A crescent is a special lune where the center of the main circle lies outside the overlapping region — i.e., the covering circle's center is close enough to exclude the main center.
The lune area is calculated using: Lune₁ = ½√((r₁+r₂+d)(r₂+d−r₁)(d+r₁−r₂)(r₁+r₂−d)) + r₁²·arccos((r₂²−r₁²−d²)/(2·r₁·d)) − r₂²·arccos((r₂²+d²−r₁²)/(2·r₂·d)). The overlap area equals πr₁² − Lune₁, and Lune₂ equals πr₂² − overlap area.
For the two circles to overlap at all, the distance d between centers must satisfy |r₁ − r₂| < d < r₁ + r₂. If d ≥ r₁ + r₂, the circles don't overlap (no lune). If d ≤ |r₁ − r₂|, one circle is entirely inside the other, which is also a special case.
Your shape is a crescent specifically when the covering circle's center is close enough to the main circle's center that the main circle's center is excluded from the overlap region. Mathematically, this happens when d < r₂ (the distance between centers is less than the covering circle's radius).
Yes. The calculator is unit-agnostic. Enter your radii and distance in any consistent unit — centimeters, meters, inches, feet, etc. — and the area result will be in the square of that unit (e.g., cm², m², in²).
The overlapping area (also called the lens or intersection area) represents the region shared by both circles. It's useful in geometry, optics, architecture, and design to understand how much of each disk is covered by the other.
Lune 2 is the crescent-shaped area of the covering circle that does not overlap with the main circle. It equals the full area of circle 2 minus the overlapping area. Together, Lune 1, the overlap, and Lune 2 account for the total combined area of both circles.