I think integration will do, I did not try it though. Here is the derivation of total surface area of the sphere by integration: http://www.mathalino.com/reviewer/derivation-formulas/derivation-formula-total-surface-area-sphere-integration.

It is interesting to note that the area of spherical zone, $A = 2\pi rh$ will become area of the sphere if $h = 2r$.

I think integration will do, I did not try it though. Here is the derivation of total surface area of the sphere by integration: http://www.mathalino.com/reviewer/derivation-formulas/derivation-formula-total-surface-area-sphere-integration.

It is interesting to note that the area of spherical zone, $A = 2\pi rh$ will become area of the sphere if $h = 2r$.