Crescent Calculator

Name: Crescent - Geometry Calculator
Author: Jürgen Kummer

Calculations at a crescent, which is formed by two equal sized circles, one cutting out the other. The intersection of both circles is a pointed oval. For crescents from two different sized circles, see lune. For the union of two circles of equal size, see two circles.
Enter circle radius and the distance of the two circle centers. Choose the number of decimal places, then click Calculate. The distance must be smaller than twice the circle radius.

Construction of a crescent from two equal sized circles.

Formulas:

h = 2 r - a

p = 2 π r

A = π r^{2} - 2 r^{2} \cdot \arccos (1 - \frac{h}{2 \cdot r}) + 2 \sqrt{r \cdot h - \frac{h^{2}}{4}} \cdot (r - \frac{h}{2})

pi:

π = 3.141592653589793...

Radius, distance, diameter and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

Another type of geometric crescent is the elliptical crescent. In this case, the two ellipses are concentric, meaning they share the same center. If circles are concentric, however, no crescent is formed. Circles of the same size completely cancel each other out, whereas circles of different sizes form an annulus.
Such crescents are axially symmetric about the line through the centers of the two original circles. This line is also the axis of rotation for rotational symmetry at an angle of 180 degrees and multiples thereof. The center of a crescent can lie outside this shape, as is the case with narrow crescents like the one in the sketch above. The crescent has two vertices, but it is not called a digon. Its two borders between both corners are circular arcs. If the two tips of the crescent are cut off in the same way by two straight lines parallel to the axis of symmetry, a curved rectangle is formed.

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