I know the formula for the area of a sphere of radio $R$, $4pi R^2$, can be derived using surface integrals, but I’m interested in how the area of a sphere of can be derived from the length of a circle of the same length. So consider a sphere of radio $R$. We can think of all the circles obtained by cutting the sphere with horizontal planes. Since the union of all these circles is the sphere and two circles do not intersect each other, it seems that the area of the sphere should be the (infinite) sum of the lengths of the circles. However, when trying to check if this works, I get the wrong result:
$$2int_0^R 2pi r dr = 2pi R^2$$
What did go wrong? I know my claim ‘the area of the sphere should be the (infinite) sum of the lengths of the circles’ must be incorrect, but I’d like to see why.
I read about the onion proof for deriving the area of a circle from its lenght and Pappus theorem to get the are of a revolution surface from the length of the generating curve, and I thought the case of the sphere and the horizontal circles would be similar, kind of an extension of Fubini’s theorem to reduce the dimension $n$ of what we want to compute and obtain it from a measure of a objects of dimension $n-1$.