I give a complete proof that the area of a circle is pi r^2, or 1/2 C r, where C is the circumference, based on the original proof by Archimedes. I also provide some historical notes and commentary on this proof and other proofs.

This is the third episode in a series called Tricky Parts of Calculus, which covers the subtle points of the subject of calculus that are rarely treated in detail in a calculus class.

Tricky Parts of Calculus, Episode 1: https://www.youtube.com/watch?v=LPpgJGgCvWc&t=1s
Tricky Parts of Calculus, Episode 2: https://www.youtube.com/watch?v=Qqfretq-18w&t=2s

3blue1brown video with circle area formula proof: https://www.youtube.com/watch?v=WUvTyaaNkzM&t=573s
Eddie Woo video with circle area formula proof: https://www.youtube.com/watch?v=Sx-5YJbszuI

Veritasium video with discussion of approximation of pi: https://www.youtube.com/watch?v=gMlf1ELvRzc&t=996s

Check out my podcast for interesting conversations about math and other topics. Especially this conversation about the historical development of mathematics: https://www.youtube.com/watch?v=vKpUW5L4fL4&t=3337s

Exercise from lecture: Prove that a point of intersection of two tangent lines to a circle is equidistant from the points of tangency.

(From Youtube)