Deriving the formula of the Surface area of a sphere My young son was asked to derive the surface area of a sphere using pure algebra. He could not get to the right formula but it seems that his reasoning is right. Please tell me what's wrong with his logic.
He reasons as follows:


*

*1.Slice a sphere into thin circles (or rings if you hollow them out)



*The sum of the circumferences of all the circles forms the surface area of the sphere.

*Since the formula of circumference is $2{\pi}R$, the sum of the circumferences would be $2{\pi}(R_1+R_2+R_3+...+R_n)$

*My son draws the radii beside each other and concludes that their sum would be equivalent to one-half of the area of the largest circle
or $({\pi}R^2)/2$. He appears to be right from his drawing.

*Substituting the sum of the radii, he comes up with a formula for the surface area of the sphere as $(R^2)*(\pi^2)$.

*He asks me what's wrong with his procedure that he cannot derive the correct formula of $4{\pi}R^2$.



An interesting proof may go through the following lines:


*

*If $S(R)$ and $V(R)$ are the surface area and volume of a sphere with radius $R$, we have $S(R)=\frac{d}{dR} V(R)$ by triangulating the surface and exploiting $S(\alpha R)=\alpha^2 S(R), V(\alpha R)=\alpha^3 V(R)$ for any $\alpha > 0$, so the problem boils down to computing $V(R)$;

*By using Cavalieri's principle and exploiting the fact that the area of a circle with radius $r$ is $\pi r^2$, we may see that the problem of computing $V(R)$ boils down to computing the area of $\{(x,y)\in\mathbb{R}^2 : 0\leq x\leq R, 0\leq y\leq x^2\}.$

*The last problem can be solved in many ways: by exploiting the optical properties of the parabola like Archimedes did, or by applying the Cavalieri's principle again and recalling that: 
$$ \frac{1}{n}\sum_{k=1}^{n}\left(\frac{k}{n}\right)^2 = \frac{(n+1)(2n+1)}{6n^2}.$$

A very short proof relying on the Gaussian integral is given by Keith Ball at page $5$ of his notorious book An Elementary Introduction To Modern Convex Geometry, but maybe it is the case to wait for your son to grow a bit :)
