TITLE: Proof without words for surface area of a sphere
QUESTION [6 upvotes]: 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!)

REPLY [14 votes]: $$\frac{dy}{ds} = \frac{r}{R} \ \implies \ 2 \pi r \cdot ds =2 \pi R \cdot dy$$
I was recently watching through 3 Blue 1 Brown videos, and learned that he did this argument, and also a second proof.