Calculate the area of a circle of radius r (distance from center to circumference) in the two-dimensional geometry that is the surface of a sphere of radius a. Show that this reduces to πr2 when r << a
Surface area of a spherical cap = 2πah = π(r2 + h2)
The Attempt at a Solution
I've thrown all the calculus I've known at this problem and couldn't crack it. I immediately realized this problem was trying to get me to calculate the area of a spherical cap in the limit where the radius of the base of the cap was much smaller than the radius of the sphere, but I tried a straightforward double integral in spherical coordinates and couldn't get it to come out right. I tried to same thing integrating over infinitesimally thin rings from the top of the sphere to the circumference of the circle. No dice. I even found the surface area for the spherical cap, which I posted above, and attempted to just show that it reduced to πr2 when r << a and couldn't even prove that much. I couldn't get a formula with r and a together. None of the handfuls of derivations I found for the surface area of a spherical cap helped me.