Circle Area by Peeling Circumference

In this short, wordless animation, we show one way to "find the area of a circle" using the method of exhaustion by unrolling 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.

For more information about this construction, see

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

To learn more about animating with manim, check out:

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Music in this video:

[oelarnes]

I know everyone knows the area of a triangle but I still wanted to hinge the left half of the triangle over to make a rectangle

[The_FBI_USA]

That's a really cool proof, I can't wait to see what's next!

[SbFH]

Amazing. I always thought of it this way.

[lolwhat]

I love this better than the other one that divides the circle until it forms a r to pi*r rectangle. I always feel like that rectangle should be r to 2*pi*r . Cant get my head around that

[KaliFissure]

I was surprised you didn't rotate one triangle to form rectangle

[Ninja]

I’ve seen another way where you cut the circle into infinitely thin sectors and then arrange them into a rectangle. The breadth will be r while the length is half the circumference, which is pi*r. Thus area is pi*r^2

[asparkdeity]

This is the video they need to show all schoolchildren when introducing pi and the area of a circle for the first time

[hitthemill]

Can you make a visualization of how area of something works, for example a square. I wanted to find something online, but didn't find anything. Would be greatly thankful.

[adwz]

Wow, great animation! u should do the surface area of a sphere and the volume of a sphere next, I have no idea how u could turn that into a visualisation but it’s something to keep u and us from being bored :D

Always remember: "What Has Been Seen Cannot Be Unseen".

[Grizzly]

This is the same video you did a month ago, but changed from pink to blue, and flipped through π/2 radians.

[SeanRhoadesChristopher]

Could this be done with a sphere, for volume? Slice it up into circles then turn those circles into triangles, then add up all the areas.

[rmela]

How do we know the layers form a perfect triangle and not something that looks like a triangle? For example how do we know for sure the sides of triangle are straight?

[wishizuk]

I always enjoy your beautiful animations! Do you use any software to create them? If so, do you mind telling us what it is?

[michaelobrien]

My instinct would be to peel into a right angle triangle!

[Zosso-]

Hey wait, are you adding lines to make an area? Or is it implied that you’re taking the limit of an annulus as its thickness approaches zero?

[kamalbouarfa]

How can the one publish his researche about a prime number théory

[jnhrtmn]

I can't step across infinity like that. It looks right, but if you never actually get there, what are we talking about? Converging or diverging, I think there is an inward and an outward infinity. Inward lets you put a bridge across it and act like it's not there, but I think that there is something more to this that people cannot see, including me. I can't let that go. In this example, you've given a width to those lines using that bridge. These videos are cool though. I studied physics, and everyone thinks angular momentum is causal, because math says so. It's not true, and NO ONE can see it. I describe a gyro using the actual "accelerations" not the pretend summation angular vector. It proves that faith in math can hide things from you.

[yiutungwong]

π = 2 in Riemann Paradox And Sphere Geometry Mathematical Systems Incorporated...

[real_kdbanman]

I really love this visualization, but the music is off-putting.

It’s so out of place that I’m actually quite hesitant to share the video.