Take one triangle (A) formed by any circle's barycenter and the two adjacent tangent points of the same circle. Then take the area of the circular sector (S) in which is traced triangle (A). The area is equal to triangle A minus three times S - A ---> A - 3(S - A). Replacing with numbers results: sqrt(3)/4 - 3[ pi/6 - sqrt(3)/4].
before i watch it, i already have an idea: a triangle between the middles of the circles is equilateral and covers 1/6 of every circle + the spot in the middle. it's sidelength is 2 times the radius of the circle so it's easy to calculate. I'll edit this when I finish watching ok im proud now xD
Every "side" of the blue shape is one sixth of the perimeter of a unit circle (because it corresponds to an angle of 60°, which is one sixth of 360°). The perimeter of a unit circle is 2*pi, one sixth of this is pi/3. Therefore the entire blue shape has perimeter pi.
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Take one triangle (A) formed by any circle's barycenter and the two adjacent tangent points of the same circle. Then take the area of the circular sector (S) in which is traced triangle (A). The area is equal to triangle A minus three times S - A ---> A - 3(S - A). Replacing with numbers results: sqrt(3)/4 - 3[ pi/6 - sqrt(3)/4].
@@P.Ripper Nice!
This is such a clean and elegant visual proof. Great job!
Thanks!
Didn't expect it to be so simple. Very nice video, gets to the point really quickly!
No need to waste time :)
25 seconds in and I already know the solution. Only time extra tutoring made me feel superior
yes I got it right
where does the number 2 come from?
Triangle base is 2 circle radii
keep going with more, pls..
More like this one or more in general? :)
😍wow!!
:)
But not so simple when all three circles are of differing radi.
before i watch it, i already have an idea:
a triangle between the middles of the circles is equilateral and covers 1/6 of every circle + the spot in the middle. it's sidelength is 2 times the radius of the circle so it's easy to calculate.
I'll edit this when I finish watching
ok im proud now xD
Way to go!
To anyone trying to solve this problem, here is a Hint...
Think of the area of a triangular piece of the circle with the vertex at the origin.
How do I calculate the perimeter?
Every "side" of the blue shape is one sixth of the perimeter of a unit circle (because it corresponds to an angle of 60°, which is one sixth of 360°). The perimeter of a unit circle is 2*pi, one sixth of this is pi/3. Therefore the entire blue shape has perimeter pi.
what is a triangle area that is not inside the circle areas
A=T-3*C
sometimes this videos make me feel i haven't studied all the way through calculus in high school 🤦♂
How so? Leave off too much detail?
what a nice result :)
😀
pretty cool :)
:) Thanks
Not purely visual proof, but nice anyway.
:)