In the book Curved Spaces by P.H.H. Wilson (pp. 35) it is defined a double lune to be the section between to maximum circles on a sphere. If $\alpha$ is the angle defining this area (see image), then the author claims that the area of a double lune is $4\alpha$.
In view of the fact that the area of $S_2$ is $4π$, it is clear that the area of the double lune is $4\alpha$.
I'm looking for a way to make this argument much more rigorous, or maybe, an alternative proof of this fact.
Let's call $f(\alpha)$ the area of the double lune. Do you agree that $f(\alpha+\beta)=f(\alpha)+f(\beta)$? If so, $f$ is a linear function of $\alpha$ and $f(\alpha)=k\alpha$, where $k$ is a constant not depending on $\alpha$. To fix $k$ we can exploit the fact that for $\alpha=\pi$ the double lune is the whole sphere, so that $f(\pi)=4\pi$ and $k=4$.