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?
Best Answer
One way to proceed is to make use of the well-known (well, it should be well-known) property of a sphere: If you inscribe a unit sphere within a right cylinder, and slice them "horizontally" (i.e., perpendicular to the axis of the cylinder) the corresponding strips of the sphere and of the cylinder have equal areas.
That this is true can be seen by examining the strips in the limit. Each strip of the sphere has smaller radius than the corresponding strip of the cylinder, by an amount equal to the cosine of the "latitude" of the strip, but by the same token, the sphere's strip is wider than the corresponding strip of the cylinder, by an amount equal to the secant of the latitude. The two factors cancel each other out.
Since the entire cylinder, neglecting the ends (which don't correspond to any portion of the sphere), has height $2$ and circumference $2\pi$, its area—and therefore the area of the sphere—is $2 \times 2\pi = 4\pi$.