Which is Larger: Four Lunes vs. Inscribed Square (visual proof)

This is a short, animated visual proof showing that the area of a square inscribed in a circle is equal to the area of the four lunes created by drawing semicircles off of the sides of the square. #manim #math #mathvideo #mathshorts #geometry #animation #theorem #pww #proofwithoutwords #visualproof #proof #area #root3 #pww​ #proofwithoutwords​ ​ #proof​ #algebra #areas #mathematics​ #mathvideo​ #mtbos

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This animation is based on an exercise from Roger Nelsen's and Claudi Alsina's wonderful book "Icons of Mathematics". You can find the book and other information about it at the links below

To learn more about animating with manim, check out:
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[mathflipped]

Great use of the Pyghagorean theorem for areas of circles!

[Kandosii]

I watched this once, then watched it a second time, enthusing out loud about *how cool this is*. It makes me want to go read up more on how we get to the final result. Bravo!

[rmela]

Algebraically: let x be radius of big circle. Then area of sq is 2x^2 since side length is sqrt(2)x

Area of circle is pi*x^2
Area of semi circles:

Area of Lunes = pi*x^2+2x^2-pi*x^2=2x^2=area of sq

[florianzellmer]

That is a very neat trick! Thank you so much for showing this.

[updown]

Wow, I didn’t know this theorem 😀 Still learning, but great visualisation bro, love this 💪👍

[spikypichu]

Wow, using the pythagorean theorem was unexpected!

[flaht]

Area square vs Area [4semi circle-4chords]
square + 4 chords vs 4 semicircle

1 circle with radius of r*sqrt2 vs 4 semicircle with radius r
2(r^2) vs 2(r^2)
they are equal

[Zosso-]

It also seems to me that, if you make the four right triangles by joining the diagonals of the square, each lune is equal to one of those right triangles, and so the square and the four lunes are equal.

[asadqasim]

TNice tutorials tutorial is so useful, I tried tons of other tutorials but tNice tutorials was the best one

[coralbelike]

hm, my algebra must’ve done me dirty somewhere. i got that the square was bigger

[pleaseenteraname]

More explanation on pythagorean theorem use here?

[kevinbihari]

Was pretty weird to calculate for you end up with 2 pi-2 pi +cube.