Circle area by exhaustion (two visual area techniques)

In this short, we show two fascinating methods of determining the area of a circle using the "method of exhaustion." Which one do you like better?

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The top animation unrolls successive circumferences of nested circular shells. In this manner, the circle area gets transformed into the area of a triangle with base 2pi times r and height r.

Here are other versions from me:
   • Circle Area by Peeling Circumference  
   • Circle area from peeling circumferenc...  

To see an alternate (and quite viral) video showing this in stop motion, check out this one from @MinutePhysics :    • Proof Without Words: The Circle  

For more information about this construction, see
https://personal.math.ubc.ca/~cass/co...

or check out this nice survey article by David Richeson from the May 2015 issue of The College Math Journal: https://doi.org/10.4169/college.math.... .

This bottom animation of a classic visual proof showing how to find the area of a circle by using more and more wedges and arranging them in a rectangle.

This proof can be traced to both Satō Moshun and Leonardo da Vinci (see Smith, David Eugene; Mikami, Yoshio (1914), A history of Japanese mathematics, https://archive.org/details/historyof..., page 130-132 and Beckmann, Petr (1976), A History of Pi, St. Martin's Griffin, page 19).

Here are other versions from me:
   • Circle Area (classic visual proof)  
   • Circle Area Derivation (visual proof)  

You can also read more about this in a great NYT article by Steven Strogatz: https://archive.nytimes.com/opinionat...

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https://manim.community