Area of dodecagon from a square!

This is a short, animated visual proof demonstrating that the area of a regular dodecagon inscribed in the unit circle has an area of exactly 3. #math​ #manim​ #visualproof​ #mathvideo​ #geometry #mathshorts​ #geometry #mtbos​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #proof​ #iteachmath #dodecagon #area #dissection

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Thanks to Matthew Scroggs for pointing me to the source.

See these videos for an alternate dissection proof of the same fact:

To learn more about animating with manim, check out:

[JohnSlack]

I think the whole world deserves to see these. They're really beautiful.

[adarshpradhan]

This is beautiful, I have tears in my eyes while seeing this magic of maths❤❤

[Scudboy]

This is why I find math to be beautiful.

[cicada.and.pomegranate]

This is a dodecagon not a dodecahedron

[ViniciusCortezao]

this manages to be beautiful, bizarre, strange, confusing, crazy and genius at the same time. I watched it 3 times to appreciate and understand

[TheDigiWorld]

Math really needs to be based on visuals rather than using million different equations to solve one problem

[abogmus]

RIP Robert Miles, never will i forget his music

[josecarloslozanopacheco]

Learning maths with the help of Robert Miles is priceless. Thank you :-)

[smileyp]

I feel like I'm learning math like the ancient Greeks!

[xjzgkke]

How do we know that the side of the triangle is equal to sqrt(3) ?

[palepoint]

The impressive thing about you that's you are always impressive 😮

[bathurstfreak]

There are 12 identical "slices of pie" of the dodecagon in the unit circle. The first slice of pie is 1 unit wide and 0.5 units high [ sin(30) = 0.5 ]. So the area of each triangle is 1 x 0.5 x 0.5 = 0.25 units so the full pie has area 12 x 0.5 = 3 units. Lovely way to demonstrate...

[MrMeszaros]

Okay - seems supersmooth.
BUT.
some steps would need a bit of explanation: like when folding the flaps in: why do they touch exactly?
How come do the blue flaps exactly fit the "star"?

Edit:
Okay, worked it out with paper and pencil. It just works. It turns out all those 30, 60, 45, 105 and 150 degrees just work out to make the top triangle a 60-60-60.
Also the bottom half-star turns to look like a square with three perfect triangles on its sides.
So in the end 150 + 60 + 90 + 60 = 360❤

[cerealkeepsyougoingeveryda]

Makes sense because a dodecagon circumscribed in a unit circle should have an area near Pi, right?

[GevinTheIdiot]

My brain has stopped working but this is beautiful

[MyHandleIsAplaceholder]

I wish you could save shorts into playlists

[balkeesshran]

Area of circle = area of square +12 small edge of (look like) semi circle is equal to 0.1415 ( From the value of π)
area of circle = 3+0.1415
= 3.1415

[davidnoll]

Cool animations, i think it’s a promising channel, but how do we know that the folds meet and make a perfect line segment in the middle?

[RNY]

Now turn a hexagon into any other polygon