Crescent Area Calculator — Lune / Moon-Shape Geometry
Find the area of a crescent (lune) shape — the region that belongs to the larger circle but not the overlapping part of the smaller inner circle — by entering the two radii and the distance between their centres.
Area of the large circle minus the overlapping region of the inner circle
Outer circle area
Intersection (lens) area of both circles
Crescent = outer area − intersection
- 1
Outer circle area
π × 5² = π × 25 = 78.5398 - 2
Lens (intersection) area
24.0826Computed from the two circular-segment arc angles using the standard lens formula. - 3
Crescent = outer area − lens area
78.5398 − 24.0826 = 54.4572
How does this calculator work?
Crescent area = area of outer circle − lens intersection of the two circles. For two circles of radii R > r with centres d apart, the lens area requires the arccos formula; when d = 0 (concentric) it simplifies to π(R² − r²). Result is in the square of your input unit.
Formula
How this is calculated
A crescent (also called a lune, from the Latin "luna" for moon) is the region inside a large circle of radius R that lies outside a smaller overlapping circle of radius r, whose centre is displaced by distance d from the large circle's centre. The crescent area equals the full outer circle area minus the lens-shaped intersection of the two circles.
When d = 0 the two circles are concentric and the "crescent" is simply an annular ring with area π(R² − r²). When the offset d is large enough that the inner circle is partially outside the outer circle (d > R − r), the intersection shrinks and the crescent area grows. Once d ≥ R + r the circles no longer overlap and the crescent would equal the full outer circle area (but a meaningful crescent no longer exists geometrically).
The lens intersection formula is derived from the two circular-segment areas that together form the lens: each segment is computed using its own arc angle (from the law of cosines) and combined to give the total overlapping area. The computation is exact — no approximations are used — so the result is as accurate as the inputs you provide.
Frequently asked questions
An annulus (ring) is the area between two concentric circles (d = 0), giving a symmetric ring with area π(R² − r²). A crescent is asymmetric — the inner circle is offset — so its area depends on how much the two circles overlap, computed via the lens formula.
It is the distance between the centres of the two circles. When d = 0 you get a concentric ring. When d = R − r the inner circle is tangent internally to the outer circle and the crescent is widest at one side and pointy at the other. As d increases beyond R − r, the intersection shrinks and the crescent looks more like a traditional moon shape.
Yes — the result is in the square of whatever unit you enter. Enter radii in centimetres and you get an area in cm²; in inches and you get in². The formula is dimensionless in structure.
TG we-Calculate Editorial Team. (2026). Crescent Area Calculator — Lune / Moon-Shape Geometry [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/crescent-area-calculator
TG we-Calculate Editorial Team. "Crescent Area Calculator — Lune / Moon-Shape Geometry." TG we-Calculate. 2026. https://we-calculate.com/calculator/crescent-area-calculator.
TG we-Calculate Editorial Team, "Crescent Area Calculator — Lune / Moon-Shape Geometry," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/crescent-area-calculator
@misc{wecalculate_crescent_area_calculator, title = {Crescent Area Calculator — Lune / Moon-Shape Geometry}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/crescent-area-calculator}}, year = {2026}, note = {TG we-Calculate} }
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