Crescent Area Calculator

What is a Crescent (Lune)?

In plane geometry, a crescent (often called a lune) is a non-convex shape bounded by two circular arcs of unequal radii that intersect at two points.

The word lune originates from the Latin word luna, meaning moon. When one circular disk partially overlaps another, the remaining non-overlapping portion forms a crescent shape.

Formulas for Crescent Area and Geometry

The geometry of a crescent formed by an outer circle of radius $R$ and an intersecting inner circle of radius $r$ separated by center distance $d$ depends on the area of the overlapping circular lens:

1. Center-to-Chord Distances

The distances from each circle center to the common intersecting chord line are:

$$d_1 = \frac{R^2 - r^2 + d^2}{2d}, \quad d_2 = d - d_1 = \frac{r^2 - R^2 + d^2}{2d}$$

2. Circular Segment Areas and Overlap Lens

Using the central angles $\alpha = 2\arccos(d_1 / R)$ and $\beta = 2\arccos(d_2 / r)$, the area of each circular segment is:

$$A_{\text{seg1}} = \frac{1}{2} R^2 (\alpha - \sin \alpha)$$ $$A_{\text{seg2}} = \frac{1}{2} r^2 (\beta - \sin \beta)$$

The total overlapping lens area is:

$$A_{\text{lens}} = A_{\text{seg1}} + A_{\text{seg2}}$$

3. Crescent Area

Subtracting the lens area from the total area of the outer circle gives the crescent area:

$$A_{\text{crescent}} = \pi R^2 - A_{\text{lens}}$$

Lune of Hippocrates

When two circular arcs share the exact same common chord with outer arc height $h_1$ and inner arc height $h_2$, the lune area is computed directly as the difference between the two circular segments:

$$A_{\text{lune}} = A_{\text{segment}}(R_1, \theta_1) - A_{\text{segment}}(R_2, \theta_2)$$

This relationship was famously studied by the ancient Greek mathematician Hippocrates of Chios (c. 470–410 BC) in his quadrature of the lune, demonstrating that curved shapes could have areas exactly equal to rectilinear polygons.

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Frequently Asked Questions

What is the difference between a crescent and a lune?

In planar geometry, the terms crescent and lune are largely synonymous. A lune specifically refers to a region bounded by two circular arcs with concave inner boundary and convex outer boundary. In astronomy and popular usage, the term crescent describes the illuminated shape of the Moon.

What happens if the distance between centers exceeds R + r?

If $d \ge R + r$, the two circles do not touch or overlap (disjoint). There is no cutout, and the resulting area is simply the full area of the outer circle ($\pi R^2$).

How do you find the perimeter of a crescent?

The perimeter is the sum of the curved lengths of the two bounding circular arcs. In two-circle intersection mode, $P = R(2\pi - \alpha) + r\beta$.

Can a lune have an area equal to a square or triangle?

Yes. Hippocrates of Chios proved that certain lunes constructed on the sides of an inscribed right isosceles triangle have areas equal to the area of the triangle itself.

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