We have two intersecting spheres $S_1$ and $S_2$ with radii $r_1$ and $r_2$. $S_1$ is centered at $(0,0,0)$, $S_2$ is centered at $(r_1, 0, 0)$. How do I calculate the surface of the intersection cap on $S_1$? I am looking for a formula of $r_1$ and $r_2$ which I can use in google sheets.
Many thanks in advance for your help.
Below is a picture representing the two spheres: the left one having radius $AB=r_1$ and the right one $BC=r_2$. Let $CD=R$ and $BD=h$, so that $AD=r_1-h$. Then by Pythagoras' theorem you have two equations: $$ \cases{ R^2+h^2=r_2^2 &\cr R^2+(r_1-h)^2=r_1^2 } $$ These can be easily solved to get $h=r_2^2/(2r_1)$ and $R=r_2\sqrt{4r_1^2-r_2^2}/(2r_1)$. Now you only need to plug these into the formula for the surface of a spherical cap: $S=2\pi R h$.
