Proof Without Words: The Circle

Prove πR² using only beads and a ruler!

I first saw this proof in an article by Russell Jay Hendel, Dowling College: http://bit.ly/LQlwQw

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Minute Physics provides an energetic and entertaining view of old and new problems in physics -- all in a minute!

Music by Nathaniel Schroeder

Thanks to Nima Doroud for contributions and to Perimeter Institute for support.
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• Category

  Science & Technology

[garnettraypaul]

If you were a teacher, you would probably be a very boring one..

[ACisking24]

For some reason I always assumed that this equation came from some stupidly complicated math I won't understand until college!!! This really blew my mind and made my day!xD As lame as that is, but considering I've been spending all weekend studying for finals, it won't take much to top that!xD

[GuitarGir464]

yes but only if the height times two pi equals the base..

[GuitarGir464]

but he is right- it's a visual proof. it doesn't matter how cool it looks this doesn't prove anything..

[supermanXL]

so i can find the area of a triangle by finding the area of a circle?

[DragcoDavid]

Actually, it has everything to do with Math AND Science. He first used objects to represent the rough area of the circle, and then used those same objects the represent the rough area of a triangle, to prove the areas are (roughly) the same, and used the area formula of a triangle to derive the area formula of a circle. That's using elementary physics, spacial continuity, and basic algebra. No, it's not exact, but it's a simple valid approximation.

[Thingie5]

hahaha this blew my mind!

[Eduardo Hirsh]

but he is right on something this "visual proof" is not correct, but helps to understand what is going on

[Eduardo Hirsh]

this is not a proof of anything, but is a fun way to understand it

[bosnianvk12]

We can if the question is why. How on the other is a total different question that deserves an answer. Very well articulated though.

[armagheddonsgw2]

For your example, sure you can make a cube out of planes, but here's the crucial bit: you need infinitely many planes *with an infinitesimal distance between them* if there's no (i.e. zero) distance between them you will still have a plane regardless of how many planes you have.

[armagheddonsgw2]

Actually, there's a massive difference between infinitesimal height and zero height. With zero height, regardless of how many objects of zero height you stack (including infinitely many) you will ALWAYS have zero height. That isn't the case with objects of infinitesimal height - the more you stack, the "taller" the stack becomes. The only problem is you need infinitely many to get anywhere.