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But WHY is a sphere's surface area four times its shadow?

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Two lovely ways of relating a sphere's surface area to a circle. Fourier socks, pi plushies, and more: Happy holidays! The first proof goes back to Greek times, due to Archimedes, who was charmed by the fact that a sphere has 2/3 the volume of a cylinder encompassing it, and 2/3 the surface area as well (if you consider the caps). Check out this video for another beautiful animation of that first proof: Special thanks to the following supporters: Calculus series: Thanks to these folks for letting me use their images at the end:
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