Say I have a sphere $x^2+y^2+z^2=a^2$ and a plane $x+y+z=b.$ How do I find the surface area of the circle cut by the sphere on the plane? I think I would use the surface integral and for graphs the surface "element" is $\sqrt{1+f_x(x,y)^2+f_y(x,y)^2}$ where $f_x$ and $f_y$ are the derivatives of $x$ and $y$ accordingly. What would $f(x,y)$ be here exactly? Or should I project this circle onto $xOy$ plane, have $f(x,y)=b-x-y$. Either way I don't know how to project the surface onto the $xOy$ plane, where it is most likely a ellipse, where then I could parameterize the domain and go on from there.
I would really like to know how to project this onto $xOy$, because I can use this for other problems.