Crescent (Lune) Area Calculator

About the Crescent (Lune) Area Calculator

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 page for Venn-diagram probability, mechanical gasket design, overlapping spotlight coverage, eclipse geometry, or any scenario where two circular regions intersect.

When This Page Helps

Use this when two overlapping circles create a leftover lune and you need the overlap, crescent, and union areas without rebuilding the circle-intersection formula by hand. It is helpful in optics, gasket design, spotlight overlap, and probability diagrams because the region percentages stay attached to the same radii and center spacing.

How to Use the Inputs

1. Enter the radius of the larger circle (R₁).
2. Enter the radius of the smaller circle (R₂).
3. Enter the distance between the centres of the two circles.
4. Select a unit (mm, cm, m, in, ft) for all measurements.
5. Read the crescent area, overlap area, union area, and percentages.
6. Use the preset buttons to explore common configurations quickly.
7. Consult the bar chart for a visual comparison of all regions.

Example Calculation

Tips & Best Practices

• If d = 0 (concentric circles), the crescent becomes an annulus: πR₁² − πR₂².
• Ensure R₁ ≥ R₂ for a standard crescent; if R₂ is larger the roles swap automatically.
• When d ≥ R₁ + R₂ the circles do not overlap and the crescent equals the full larger circle area.
• For eclipse modelling, set R₁ to the Sun’s apparent radius and R₂ to the Moon’s.

When To Use This Calculator

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. Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.

How To Check The Result

Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.

Practical Notes

Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.

Frequently Asked Questions

• What is a crescent in geometry?

  A crescent (lune) is the area of one circle that does not overlap with another circle when the two circles partially intersect.

• How is crescent area different from an annulus?

  An annulus is the ring between two concentric circles (d = 0). A crescent occurs when the circles share different centres and partially overlap.

• What happens when one circle is inside the other?

  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.

• Can the two radii be equal?

  Yes. Two equal circles produce a symmetric lens-shaped overlap, and the crescent on either side is identical.

• What is the overlap formula based on?

  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.

• Is a crescent the same as a Hippocrates lune?

  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.