About

A crescent (or lune) is the region between two overlapping circles where the smaller circle's area is subtracted from the larger. Calculating its area is not trivial. Unlike a simple annulus where circles share a center, a crescent allows an offset distance d between centers. The computation requires resolving the circle-circle intersection area via inverse trigonometric functions and the Heron-like radical term √(−d + R₁ + R₂)(d + R₁ − R₂)(d − R₁ + R₂)(d + R₁ + R₂). Getting this wrong in optical lens design, architectural arches, or CNC cutting paths means wasted material and failed tolerances. This calculator handles all geometric cases: concentric circles (d = 0), fully interior placement, partial overlap, and tangential contact.

The tool assumes both circles lie in the same Euclidean plane. Results approximate to floating-point precision (~15 significant digits). For non-circular crescents (elliptical lunes), numerical integration would be required. Pro tip: when d = 0, the crescent reduces to an annulus with area π(R₁² − R₂²).

Formulas

The crescent area is defined as the area of the outer circle minus the intersection area of the two circles. For partial overlap:

A_crescent = πR₁² − A_int

The circle-circle intersection area A_int for two circles of radii R₁ and R₂ with center distance d:

A_int = R₁² arccos(d² + R₁² − R₂²2dR₁) + R₂² arccos(d² + R₂² − R₁²2dR₂) − 12√S

Where the radical term S uses the four-factor expression:

S = (−d + R₁ + R₂)(d + R₁ − R₂)(d − R₁ + R₂)(d + R₁ + R₂)

Variable legend: R₁ = radius of the outer (larger) circle. R₂ = radius of the inner (smaller) circle. d = distance between the two circle centers. A_int = area of the overlapping (lens/vesica) region. A_crescent = area of the resulting crescent (lune). When the inner circle is fully contained (d + R₂ ≤ R₁), the intersection equals the full inner circle: A_int = πR₂², so A_crescent = π(R₁² − R₂²).

Reference Data

Geometric Case	Condition	Crescent Area Formula	Notes
Concentric (Annulus)	d = 0	π(R₁² − R₂²)	Simplest case, symmetric ring
Inner circle fully inside	d + R₂ ≤ R₁	πR₁² − πR₂²	No intersection boundary needed
Partial overlap	R₁ − R₂ < d < R₁ + R₂	πR₁² − A_int	General case, uses full intersection formula
Externally tangent	d = R₁ + R₂	πR₁²	Circles touch at one point, no overlap
No overlap	d > R₁ + R₂	πR₁²	Inner circle doesn't subtract anything
Internally tangent	d = R₁ − R₂	πR₁² − πR₂²	Boundary case of full containment
Equal radii, offset	R₁ = R₂, d > 0	Symmetric lune pair	Two equal crescents formed
Hippocrates' Lune	Special ratio	Equals related triangle area	One of the earliest quadrature results (5th c. BCE)
Half-moon	R₂ = R₁, d = 0	0	Circles coincide, no crescent
Thin crescent	R₂ ≈ R₁, small d	≈ 2πRd	Linear approximation for thin crescents
Lunar phase (astronomy)	Projected sphere geometry	Spherical crescent formula differs	Flat formula is a 2D approximation only
Lens (vesica region)	Overlap of two circles	A_int directly	The intersection itself, not the crescent
Optical lens cross-section	Biconvex lens profile	Sum of two crescents	Used in optics manufacturing tolerances
Gear tooth profile	Involute approximation	Crescent-like region	Approximated by circular arc crescents
Cam lobe area	Eccentric circles	Crescent integral	Mechanical engineering application

Frequently Asked Questions

When the center distance d is large enough that R₂ protrudes beyond R₁, the crescent area increases because the intersection area shrinks. The calculator handles this correctly using the general intersection formula. The crescent approaches πR₁² as the circles separate entirely.

At d = 0, the crescent is a symmetric annulus with area π(R₁² − R₂²). As d increases from zero (inner circle still fully contained), the crescent area stays constant at π(R₁² − R₂²). Once d exceeds R₁ − R₂, partial overlap begins and the crescent area grows because less of the inner circle overlaps with the outer.

Yes. When R₁ = R₂ and d > 0, two symmetric lunes are formed (one on each side). The calculator computes the crescent as the outer circle minus the intersection. At d = 0 with equal radii, the crescent area is zero because the circles coincide.

The maximum crescent area is πR₁², achieved when the circles do not overlap at all (d ≥ R₁ + R₂). In this case, the inner circle subtracts nothing from the outer circle. The minimum crescent area (for R₂ < R₁) is π(R₁² − R₂²), occurring when the inner circle is fully contained.

No. The lunar crescent is a projection of illumination on a sphere, not a 2D circle-circle subtraction. The visible Moon phase involves spherical geometry and the angle between the Sun, Moon, and observer. This calculator handles planar Euclidean crescents only. For lunar phase area, you would need to integrate over the illuminated spherical cap projected onto a disk.

Near these boundary values, the product inside the square root approaches zero, which can cause floating-point precision issues. The calculator checks for these edge cases explicitly: if the product S is negative due to rounding (theoretically impossible for valid inputs), it is clamped to 0. This prevents NaN results. Precision is maintained to approximately 10 decimal places for typical inputs.