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

2018-12-02

[public] 4.48M views, 147K likes, 1.62K dislikes audio only

3Blue1Brown

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The formula is no mere coincidence.

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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:

https://youtu.be/KZJw0AYn6_k

Calculus series:

http://3b1b.co/calculus

Thanks to these viewers for their contributions to translations

German: @Dat-Pudding

Hebrew: Omer Tuchfeld

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If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.

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0:00 - High-level idea

2:23 - The details

9:12 - Limit to a smooth surface

11:20 - The second proof

15:15 - A more general shadow fact.