Fig 1.
Using n lines to cut one half of the circle into n parts.The distance between two lines is d.The lines cut the x-axis at x1,x2,......,xn respectively.
1. Prove the surface area of sphere
by CALCULUS:
First ,put a circle with radius
r on a graph , and let the equation of it be:
y= f(x) or x^2 + y^2 = r^2 as shown in Fig
1.
Fig 1.
Using n lines to cut one half of
the circle into n parts.The distance between two lines is d.The lines cut
the x-axis at x1,x2,......,xn respectively.
So the surface area of the sphere
should be:
lim 
2 Pi f(xi)
[d^2 + (f(x+d)-f(d))^2]^0.5
d-->o
lim
2 Pi f(xi) [1 + (xi/(r^2-xi^2))^2] ^0.5 d
d-->o
= 
= 2Pi r^2
So, the surface area of a hemisphere
is 2 Pi r^2 and the surface area of a sphere is
4 Pi r^2