Derivation (Proof) of Surface Area of a Sphere

Consider a sphere of radius $r$. We would like to find the surface area of this sphere.

We know that the volume $V$ of the sphere of radius $r$ is $V=\frac{4}{3}\pi r^3$.

Calculating the derivative of $V$ with respect to $r$,

$\frac{dV}{dr}=\frac{d}{dr}(\frac{4}{3}\pi r^3)=4\pi r^2$.

However, notice that the additional volume $\Delta V$ can be approximated by "unraveling" the outer shell of the sphere. Thus, the extra volume is roughly the surface area of the sphere $A$ times the additional radius $\Delta r$. This approximation becomes more and more accurate as $\Delta r$ approaches zero, and it becomes exact in the limiting case.

Thus, $A dr=dV$, which means that $A=\frac{dV}{dr}$, so $A=4\pi r^2$.

This means that the surface area of a sphere is $4\pi$ times the radius squared.