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

The formula is no mere coincidence.
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Discussion on Reddit: https://www.reddit.com/r/3Blue1Brown/comments/a2gqo0/but_why_is_a_spheres_surface_area_four_times_its/

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 folks for letting me use their images at the end:
https://www.youtube.com/user/vlogbrothers
https://www.youtube.com/user/physicswoman
https://www.youtube.com/user/Vsauce
https://www.youtube.com/user/onemeeeliondollars

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These animations are largely made using manim, a scrappy open source python library: https://github.com/3b1b/manim

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.

Music by Vincent Rubinetti.
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https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown

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If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.

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Видео But why is a sphere's surface area four times its shadow? канала 3Blue1Brown