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 $A_{C} = \frac{\pi rw}{2}$, where $w$ is the shift from center $'X'$ in blue to $'X'$ in red.
Details:
$$A_C = \frac{A_{elipse} - A_{circle}}{2} = \frac{[\pi r^2 + \pi r w] - \pi r^2}{2}$$


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:
$$A_{\text{crescent}}=\pi r^2-(2A_{\text{sector }EAF}-2A_{\triangle AEF})$$
This is because $A_{\text{sector }EAF}=A_{\text{sector }ECF}$, 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 $A=\frac12ab\sin C$. Then the area of the crescent can be re-written as:
$$A=\pi r^2-\alpha r^2+r^2\sin\alpha\\
\implies A=r^2(\pi-\alpha+\sin \alpha)$$
