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

The formula is no mere coincidence. Help fund future projects: An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: Discussion on Reddit: 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: Calculus series: Thanks to these viewers for their contributions to translations German: @Dat-Pudding Hebrew: Omer Tuchfeld ------------------ These animations are largely made using manim, a scrappy open-source python library: 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. Download the music on Bandcamp: Stream the music on Spotify: 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. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: Various social media stuffs: Website: Twitter: Reddit: Instagram: Patreon: Facebook: 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.