I love the book Proofs Without Words by Roger B. Nelsen. One of the proofs I liked the most was this: Area under one arch of a cycloid is 3 times the area of the wheel that traces it. You break the cycloid in to three parts and show that each part has an area equal to that of the wheel.


I have always wondered if there is a similar proof for why the surface area of a sphere is equal to the area of a circle with radius twice that of the sphere. Is there a nice way to see this?


I would also like to know if one can show without doing any calculus that the length of the arch of the cycloid is 8 times the radius of the wheel. (Note: $\pi$ does not show up here, so this sounds tricky!)



Proof without words for surface area of a sphere
It seems that you are more or less asking for a proof-without-words of Archimedes's theorem equating the area on a sphere between two horizontal slices with the area of the circumscribed cylinder between the same two slices. But this is a notoriously non-obvious theorem, so I'll be very surprised if such a proof exists.


Here's why I say that proving the $4\pi r^2$ formula is more or less the same as proving Archimedes's theorem: The theorem is clearly approximately true for a very thin slice symmetric around the equator. As you move from the equator to a pole, taking slices of equal width, it would be at least mildly suprising if the corresponding spherical surface areas did not change monotonically. If they decrease, then the total area of the sphere is less than that of the cylinder; if they increase, then the total area of the sphere is more than that of the cylinder. So (at least given the expectation of monotonicity), having them all equal (i.e. the full strength of Archimedes's theorm) is equivalent to the sphere having the same surface areas as the cylinder, which in turn is clearly $4\pi r^2$.


