Is the following Area of Crescent all right?

In the figure below. There are two overlapping circles and the area of Crescent in Red that I have found is $A_C = fracpi rw2$, where $w$ is the shift from center $'X'$ in blue to $'X'$ in red.

Details:
$$A_C = fracA_elipse - A_circle2 = frac[pi r^2 + pi r w] - pi r^22$$

2 Answers
2

WLOG, assume both centers lie on the $x$-axis. You can use this diagram afterwards:

Since the area of the circle is $A=pi r^2$, then the area of the crescent should be:
$$A_textcrescent=pi r^2-(2A_textsector EAF-2A_triangle AEF)$$

This is because $A_textsector EAF=A_textsector ECF$, and so does their corresponding triangle. Since the area of a sector is $A=frac12r^2 theta$, with $theta$ in radians, and the area of the triangle is $A=frac12absin C$. Then the area of the crescent can be re-written as:
$$A=pi r^2-alpha r^2+r^2sinalpha\
implies A=r^2(pi-alpha+sin alpha)$$

Note that for $w=r$ we obtain

$$A_C=fracpi r^22$$

which seems to be wrong.

Moreover how does the area for the ellipse come in the derivation?

For the general formula refer to Circle overlapping.

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