I help mentor some really young, bright kids in mathematics. We were looking at geometric properties of various shapes, and one of the kids noted that the surface area of a sphere $S = 4\pi r^2$ contains the equation for the area of a circle $A = \pi r^2$.
She was a bit confused why the factor of $4$ was mysteriously there. I told her I'd get back to her.
I know how to prove the formula using calculus, but I spent a long time trying to find an elementary way of doing it.
Does anyone know of a way of proving the first equation using almost no advanced mathematics$^1$? This seems unlikely, so as a separate question, does anyone know of a good visualization to show the relation between $S$ and $A$?
The naive approach of taking four circles and showing you can "place them" on a sphere is clearly wrong (you can't just place four circles on a sphere), but I'm not sure what the alternative is.
$^1$These kids have a working knowledge of variable manipulation, basic geometry, and I guess combinatorics?