Calculate the area of a crescent or lune formed by two overlapping circles. Enter radii and distance between centers to find crescent area, overlap area, and union area.
A crescent — also called a lune — is the region of a larger circle that is not covered by a smaller overlapping circle. Crescents appear throughout science, engineering, art, and everyday life: from the thin sliver of the waxing Moon to the cross-sections of piping and optical apertures.
This calculator uses the exact circle–circle intersection formula to determine the overlap (lens-shaped) area between two circles of radii R₁ and R₂ whose centres are separated by a distance d. The crescent area is then the area of the larger circle minus the overlap. You also get the union area, individual circle areas, and the percentage each region represents.
The underlying mathematics relies on the law of cosines and circular segment areas. When the two circles do not touch (d ≥ R₁ + R₂), the overlap is zero and the crescent equals the full larger circle. When one circle is entirely inside the other (d + R₂ ≤ R₁), the overlap equals the smaller circle, and the crescent is the classic annular shape. For all other configurations an integral-derived closed-form expression computes the exact intersection.
Use this tool for Venn-diagram probability, mechanical gasket design, overlapping spotlight coverage, eclipse geometry, or any scenario where two circular regions intersect.
This crescent (lune) area calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Larger circle radius (R₁), Smaller circle radius (R₂), Distance between centers (d) and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Overlap = r₁²·cos⁻¹((d²+r₁²−r₂²)/(2dr₁)) + r₂²·cos⁻¹((d²+r₂²−r₁²)/(2dr₂)) − ½√[(−d+r₁+r₂)(d+r₁−r₂)(d−r₁+r₂)(d+r₁+r₂)] Crescent = πR₁² − Overlap Union = πR₁² + πR₂² − Overlap
Result: For largercircle=5, smallercircle=10, distancebetween=15, the tool returns the solved crescent (lune) area outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in crescent (lune) area formulas and reports derived values, checks, and classifications automatically.
This page is tailored to crescent (lune) area, with outputs tied directly to the form fields (Larger circle radius (R₁), Smaller circle radius (R₂), Distance between centers (d)). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
A crescent (lune) is the area of one circle that does not overlap with another circle when the two circles partially intersect. Use this as a practical reminder before finalizing the result.
An annulus is the ring between two concentric circles (d = 0). A crescent occurs when the circles share different centres and partially overlap.
The overlap equals the smaller circle\u2019s area and the crescent is the larger circle area minus the smaller circle area, similar to an annulus. Keep this note short and outcome-focused for reuse.
Yes. Two equal circles produce a symmetric lens-shaped overlap, and the crescent on either side is identical.
It uses the areas of two circular segments formed by the chord of intersection, derived from the law of cosines and the segment area formula A = r²(θ − sin θ)/2. Apply this check where your workflow is most sensitive.
A Hippocrates lune is a specific crescent whose area equals a related rectilinear figure — a famous result in Greek mathematics. The general crescent calculator handles any configuration.