Crescent Area Calculator (circle with an overlapping circle cut away)

Understanding the crescent area calculator — concept and principle

This calculator finds the area of a crescent, the part of a circle left after a second, overlapping circle is cut away. It also gives the overlap (lens) area, the area of the full circle and the share that remains. Use it for geometry problems, logo and moon-shape design, cutting fabric or sheet material.

How it is calculated

Crescent area = πR² − lens area

For partly overlapping circles (|R − r| < d < R + r):

Lens = r² arccos((d² + r² − R²) ÷ (2dr)) + R² arccos((d² + R² − r²) ÷ (2dR)) − ½ √((−d + r + R)(d + r − R)(d − r + R)(d + r + R))

• R = radius of the outer circle, r = radius of the cutting circle, d = distance between their centres.
• If d ≥ R + r the circles do not touch and the whole circle remains.
• If the cutting circle lies fully inside, the result is a ring with a hole: πR² − πr². If it covers the outer circle, nothing is left.

The principle

The overlap of two discs is a lens made of two circular segments, one from each circle. Each segment is a sector minus a triangle. The arccos terms give the two sector areas (angle × radius² ÷ 2, written with half-angles), and the square-root term is the area of the kite formed by the two centres and the two intersection points, found with Heron's formula. Subtracting the lens from the outer circle leaves the crescent.

Worked example

Default inputs: R = 10 cm, r = 8 cm, d = 6 cm.

1. Outer circle = π × 10² = 314.159 cm².
2. First arccos argument = (36 + 64 − 100) ÷ 96 = 0, so arccos = π/2 and the term is 64 × 1.5708 = 100.53.
3. Second argument = (36 + 100 − 64) ÷ 120 = 0.6, arccos 0.6 = 0.9273, so the term is 100 × 0.9273 = 92.73.
4. Root term = ½ √(12 × 4 × 8 × 24) = ½ √9,216 = 48; lens = 100.53 + 92.73 − 48 = 145.26 cm².
5. Crescent = 314.159 − 145.26 = 168.899 cm², which is 53.76 % of the circle. Shape: Crescent.

How to read the result

The share left tells you how "thin" the crescent is. As d shrinks toward R − r, the cut moves inside and the shape turns into a ring; as d grows toward R + r, the overlap vanishes. Areas come out in the square of the length unit you entered.

Common mistakes

• Entering diameters instead of radii.
• Measuring d between the circle edges instead of the centres.
• Using degrees inside the arccos formula by hand; the formula needs radians.
• Assuming the crescent is half the circle.

Frequently asked questions

How do you find the area of a crescent moon shape?

Subtract the overlap of the two circles from the area of the outer circle.

What is a lens in geometry?

The region shared by two overlapping circles.

What happens if the cutting circle is entirely inside?

You get a ring-like shape with area πR² − πr².

Sources: E. W. Weisstein, "Circle-Circle Intersection", Wolfram MathWorld, eq. (15)