Crescent Area Calculator - Lune, Lens, and Overlap Areas
crescent area calculator returns the lune, lens, and overlap areas for two overlapping circles from r1, r2, and the center distance d.
Crescent Area Calculator
Results
What Is the Crescent Area Calculator?
The crescent area calculator returns three areas for two overlapping circles in one step: the lune of circle 1, the lune of circle 2, and the lens where the two circles meet. Use it for homework on lune and lens areas, for a crescent in a logo or stained-glass layout, and for any design problem where one circular feature partly covers another.
- • Geometry homework on lunes and lenses: Solve textbook problems that ask for the area of a lune, a lens, or the crescent portion of a lune.
- • Logo, stained-glass, and mosaic layouts: Compute the area of a moon-shaped panel cut from one disk by another, and check the lens for material estimates.
- • Engineering and machining overlaps: Estimate the area where two circular features overlap on a part, such as a hole partly covering a boss.
- • Astronomy and visual demonstrations: Model the bright crescent of a partially lit disk and the overlap between two disks for teaching lune, lens, and crescent differences.
A crescent is a lune that also excludes the center of the original disk, which happens when d is smaller than the opposite radius. The calculator returns the full lune area for each disk and a live Crescent or Not a crescent label beside each row.
When the same two radii also need the full disk area, circumference, and diameter, the Circle Calculator returns those values from a single radius input.
How the Crescent Area Calculator Works
The calculator applies the closed-form lune area formula attributed to Eric Weisstein's work on Wolfram MathWorld, then subtracts the first lune from pi * r1^2 to get the lens overlap and from pi * r2^2 to get the second lune. All three areas trace back to the same r1, r2, and d inputs.
- r1: Radius of the first (covering) circle, in the chosen linear unit.
- r2: Radius of the second (covered) circle, in the same linear unit as r1.
- d: Center-to-center distance. Must satisfy |r1 - r2| < d < r1 + r2.
- lune1: Area of the curved region of circle 1 that lies outside circle 2 (primary output, square units).
- lune2: Area of the curved region of circle 2 that lies outside circle 1, computed as pi * r2^2 minus the lens overlap.
- overlap: Area of the lens (convex overlap of both disks), computed as pi * r1^2 minus lune1.
The formula has three parts: a Heron-style square root of four product terms in r1, r2, and d, plus r1 squared times an arccos term, minus r2 squared times a second arccos term.
Arccos arguments can drift outside [-1, 1] from rounding, so the calculator clamps each argument to that interval before calling arccos and rejects any input that fails the overlap test.
The three results always sum to the union area of the two disks: pi * r1^2 + pi * r2^2 minus the lens overlap. As a quick check, lune1 + lens equals pi * r1^2 and lune2 + lens equals pi * r2^2.
Worked example: r1 = 4, r2 = 6, d = 5
Enter r1 = 4, r2 = 6, d = 5 with the same linear unit on all three inputs.
Radicand = 15 * 7 * 3 * 5 = 1575, sqrt term = 0.5 * sqrt(1575) = 19.84. acos(-0.125) = 1.6961, acos(0.75) = 0.7227. lune1 = 19.84 + 16 * 1.6961 - 36 * 0.7227 = 20.96.
Lune 1 = 20.96 square units, lens overlap = pi * 16 - 20.96 = 29.30 square units, lune 2 = pi * 36 - 29.30 = 83.79 square units.
Lune 1 is a crescent because d = 5 is smaller than r2 = 6. Lune 2 is not a crescent because d = 5 is not smaller than r1 = 4.
According to Wolfram MathWorld, the area of a lune of two circles with radii r1 and r2 and center distance d equals a square root of four product terms plus two arccos terms.
For the flat shape that is not two overlapping circles, the Area Calculator returns the area of rectangles, triangles, and other common forms.
Key Concepts Explained
These four terms describe the curved regions that two overlapping circles can form. The lune is the building block for the crescent and the lens.
Lune
The concave-convex region of one disk that lies outside the other. Each disk has its own lune, and the two lunes plus the lens always equal the union area (pi * r1^2 + pi * r2^2 minus the lens overlap).
Crescent
A lune that does not contain the original disk's center. A lune becomes a crescent when d is smaller than the other circle's radius; otherwise the lune wraps around the original center.
Lens
The convex-convex region where the two disks overlap. It is symmetric across the line connecting the two centers, and its area equals pi * r1^2 minus the first lune.
Center distance
The straight distance d between the two disk centers. A small d gives a large lens and tiny lunes; a d close to r1 + r2 gives a thin sliver of a lens.
If r1 equals r2, the two lunes are equal in area and the lens is centered on the perpendicular bisector of the segment between the two centers.
The boundary of each lune is a circular arc, and the Arc Length Calculator returns the arc length from a radius and a central angle.
How to Use the Crescent Area Calculator
Use the calculator with one linear unit for the two radii and the center distance, and keep d strictly between |r1 - r2| and r1 + r2 so the disks overlap in a lens.
- 1 Enter the radius of circle 1: Type the radius of the first (covering) circle. Pick a unit and stay with it for r2, d, and the result.
- 2 Enter the radius of circle 2: Type the radius of the second (covered) circle. For equal circles, enter the same number for r1 and r2.
- 3 Enter the center-to-center distance: Type the distance between the two centers. The value must lie strictly between |r1 - r2| and r1 + r2.
- 4 Read the area of lune 1: Use lune 1 as the curved area of circle 1 outside circle 2. Lune 1 is also a crescent whenever d is smaller than r2, regardless of which disk is larger.
- 5 Read the lens overlap area: Use the lens area for the region inside both circles, such as the visible area where a hole and a boss overlap on a part.
- 6 Read the area of lune 2: Use lune 2 as the curved area of circle 2 outside circle 1. Lune 2 is also a crescent whenever d is smaller than r1, regardless of which disk is larger.
Suppose you are designing a stained-glass window where a 4 cm radius disk partly covers a 6 cm radius disk, with the centers 5 cm apart. Lune 1 is 20.96 square centimeters and is labeled a crescent because d = 5 is less than r2 = 6. The lens is 29.30 square centimeters, and lune 2 is 83.79 square centimeters and is labeled not a crescent because d = 5 is not less than r1 = 4.
If the design moves from two overlapping circles to a flat polygon outline, the Polygon Area Calculator covers regular polygons, irregular polygons, and triangles.
Benefits of Using the Crescent Area Calculator
Reading the two lune areas and the lens overlap in one step turns three plane-geometry inputs into a complete picture of a two-circle overlap.
- • Three results from three inputs: Get the lune for each disk and the lens overlap in one step, with no re-entry or extra step.
- • Dynamic crescent flag for each lune: A live Crescent or Not a crescent label sits next to each lune, recomputed from d versus the opposite radius.
- • Single linear unit across all results: Every area uses the same square unit as the linear input, so the lens and the two lunes can be compared without an extra conversion step.
- • Numeric guards on arccos: Arccos arguments are clamped to [-1, 1] before the call, so a near-edge input still returns a valid result.
All three results come from the same three inputs, so the calculator also works as a formula checker. Enter a textbook problem's r1, r2, and d, then read the lune and lens areas to confirm a hand calculation.
The straight chord across the lens boundary complements the lens area, and the Chord Length Calculator gives the chord length plus the segment area for the same r1, r2, and d.
Factors That Affect Your Results
The formula is fixed, but the choice of input values changes which lune is the crescent and how big the lens ends up.
Overlap test on d
d must be strictly greater than |r1 - r2| and strictly less than r1 + r2. Outside that range the disks either do not overlap at all, or one disk sits fully inside the other with no lune to measure.
Which lune becomes a crescent
Lune 1 is a crescent when d is smaller than r2, and lune 2 is a crescent when d is smaller than r1. A lune is a crescent any time the center of its own disk lies inside the other disk; the radius comparison is not part of the rule, and the result panel updates its label as the inputs change.
How close the centers are
When d is close to 0 the lens area approaches pi times the smaller disk squared and each lune is tiny. When d is close to r1 + r2 the lens becomes a thin sliver.
Arccos argument clamp
Arccos is only defined on [-1, 1]. The calculator clamps each argument to that interval before the call, so a near-edge input still returns a finite area instead of NaN.
- • The formula treats both circles as flat 2D disks. For two spheres in 3D, use spherical-cap or spherical-overlap formulas; the planar lune area does not equal the curved lune on a sphere.
- • The result is exact for an ideal two-disk overlap, so treat it as a working estimate. Rounding to two decimal places can differ by a few hundredths from a hand calculation that rounds after every step.
According to Wolfram MathWorld, the lens area is pi times r1 squared minus the area of the corresponding lune.
According to Wikipedia, a lune is a concave-convex region bounded by two circular arcs, and a crescent is a lune that does not contain the original disk's center.
For flooring or paint jobs that need the full disk area in square feet rather than the lune and lens, the Square Footage of a Circle Calculator turns the same radius into a square-footage result.
Frequently Asked Questions
Q: What is the formula for the area of a crescent?
A: A crescent is a lune that does not contain the original disk's center, and the lune area is given by 0.5 * sqrt((r1+r2+d)(r2+d-r1)(d+r1-r2)(r1+r2-d)) plus r1^2 arccos((r2^2-r1^2-d^2)/(2 r1 d)) minus r2^2 arccos((r2^2+d^2-r1^2)/(2 r2 d)), where r1 and r2 are the two radii and d is the center-to-center distance.
Q: How do I find the area of a lune?
A: Apply the closed-form lune formula to r1, r2, and d, then check that d lies strictly between |r1 - r2| and r1 + r2 so the disks actually overlap. The crescent area calculator does the same calculation and also tells you whether the lune is a crescent.
Q: What is the difference between a crescent, a lens, and a lune?
A: A lune is the concave-convex region of one disk that lies outside the other. A lens is the convex region shared by both disks. A crescent is a lune that also excludes the original disk's center, which happens when d is smaller than the opposite radius.
Q: How do I calculate the area where two circles overlap?
A: The lens overlap equals pi times r1 squared minus the area of the first lune. Once you know lune1, the overlap follows directly, and lune2 equals pi times r2 squared minus the same overlap.
Q: What is the area of a lens formed by two circles?
A: The lens is pi * r1^2 - lune1 (or equivalently pi * r2^2 - lune2). For two equal circles of radius r with center distance d, the lens area is 2 * r^2 * arccos(d / (2r)) - 0.5 * d * sqrt(4 r^2 - d^2).
Q: When does a lune become a crescent?
A: A lune becomes a crescent when the center-to-center distance d is smaller than the radius of the other disk. In that case the original disk's center lies outside the lune, and the lune takes the moon-like crescent shape that gives this calculator its name.