Elliptical Crescent Calculator

Calculations at an elliptical crescent. This is a crescent, which isnt't based on two circles, but on two ellipses. Those two ellipsed have a common semi-axis of equal length. The other semi-axes, which are perpendicular to it, have different lengths. The elliptical crescent is formed by the area between each pair of semi-ellipses.
Enter the common semi-axis and the two different semi-axes of the two underlying ellipses. Choose the number of decimal places, then click Calculate.

Formulas:
p ≈ { π * (a+b) * [ 1 + 3λ² / (10+√4-3λ² ) ] + π * (a+c) * [ 1 + 3μ² / (10+√4-3μ² ) ] } / 2
λ = ( a - b ) / ( a + b )
μ = ( a - c ) / ( a + c )

A = π * a * ( b - c ) / 2

Kreiszahl pi:
π = 3.141592653589793...

Semi axes and circumference have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

Unlike a crescent, which is formed with two circles, the two ellipses of an elliptical crescent share a common center. If these ellipses were circles, they would be the same size and located at the same point; subtracting one from the other would leave nothing behind. However, the shape of an elliptical crescent results from the difference in their semi-axes. It doesn't matter whether the common semi-axis of the two ellipses is the major or minor one. In the sketch above, it is the major semi-axis. Of the other two semi-axes, the one with the longer length is called the outer semi-axis, and the one with the shorter length is called the inner semi-axis. The circumference is calculated using Ramanujan's second approximation, just like at a standard ellipse. This is the arithmetic mean of the circumferences of both underlying ellipses.