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?

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:

For more information about this construction, see

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.

Here are other versions from me:

#math #manim #visualproof #proofwithoutwords #circle #circlearea #archimedes #radius #area #areaofcircle #pi #piday #shorts #circle #archimedes #infinite #methodofexhaustion

To learn more about animating with manim, check out:

[erinbriggs]

The top one is more unique. Had not seen it displayed that way before, but as a limit, they are both spot-on accurate.

[codatheseus]

Method of exhaustion sounds like you're tired of explaining the same thing in so many ways

[Alhoshka]

This man didn't just square the circle. He triangled it too!

[totallynotpaul]

It was a day after the AP test in my calc class, so the teacher was just teaching random stuff cause it didn't really matter. At some point, one student said calculus is completely unrelated to all other math, so my teacher just started deriving a bunch of random math formulas we knew using calculus for the next week or so. It was actually pretty interesting.

[arenmee]

This is probably the answer to why pizzas are shaped like a circle but are placed inside square boxes while they're eaten as triangles.

[illumencouk]

I found that presenting both together like this, increased their value to three.

[ET_AYY_LMAO]

The bottom one is most intuitive because most can relate to calculating the area of a rectangle.

[itllmakesense]

Bottom Is Better For Me

But Both Ways Are Unique And Both Usable

[jefejeffwell]

The triangle circle limit is awesome. Well, they’re both awesome, the circle is just more awesome

[robertlaws]

I like the top one. I never saw it so well explained.

[ordiv]

It's Mindblowing! Thanks for your contribution to education!

[Zubigri]

Second is kinda more easy and satisfying

[islandwills]

I dont think anyone ever explained how those formula worked or the reasoning behind them. I really wish they did because the way that i learn is greatly aided by knowing the why.

[kpfields]

I like the top one more. For some reason, it makes more sense to me.

[christianaquilina]

Yes... I like this representation of both shown together

[electron_]

The lower one is learned and well known, the upper one is completely new for.me and totally excellent. Great video.

[donaldhobson]

The top one would make a really cool book.

[franciscomoralesmorillas]

Que animación tan perfecta y que bien explicado. Gran trabajo!❤

[KaliFissure]

I still love that the circumference of a circle radius Pi is 2 Pi squared.
And the area Pi cubed.

[alexbeecroft]

I've seen the wedge approach many times but this was my first time seeing the triangular approach. I really like it!