TITLE: Can the Surface Area of a Sphere be found without using Integration?
QUESTION [15 upvotes]: When we were in school they told us that the Surface Area of a sphere = $4\pi r^2$
Now, when I try to derive it using only high school level mathematics, I am unable to do so. Please help.

REPLY [3 votes]: The volume of a rotated graph of a function $f$ around the $x$-axis segment $[a,b]$ is $V=\pi \int_a^b f^2(x)dx$.
You could describe the unit sphere as the rotated graph of $f(x)=\sqrt{1-x^2}$ around $[-1,1]$. Then the volume of the sphere is $V=4\pi /3$. By a scaling argument, the general formula can be derived.
For the area of the sphere, my primary school teacher said that it could be thought of a pyramid with base the surface of the sphere. Then applying the volume formula: $Volume=(Base Area)\times Height / 3$ it follows that
$$ 4\pi R^3/3=A\times R/3$$ Then $A= 4\pi R^2$. The last part is not rigorous at all, but it works. :)