How do you derive the formula for the surface area of a sphere?
Archimedes did it by slicing the sphere as you did, and proving by geometry that the area of each slice was the same as the area of a cylinder radius r wrapped around the sphere. That is fairly easy using trig (but harder for Archmedes, because the Greeks never invented trigonometry). The area of the slice of cylinder is 2 pi r dx. The slice of sphere has radius r cos theta (where theta is the “latitude” of the slice above the “equator” of the sphere ). The surface of the spherical slice is tilted at an angle, and its length is dx/cos theta. So (2 pi r cos theta) (dx / cos theta) = 2 pi r dx and the slices of sphere and cylinder have equal areas. The area of the whole cylinder is (2 pi r) (2 r) = 4 pi r^2.
