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Thread: Three equal circle in the big circle

  1. #1
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    Three equal circle in the big circle

    Hello guys,

    In the big circle is three little equal circle.

    can you tell me what is area of blue figure? if big circle radius is 2 + √3

    Three equal circle in the big circle-circle.png
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  2. #2
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    Re: Three equal circle in the big circle

    Let $R$ be the radius of the large circle and $r$ be the radius of any of the 3 smaller circles.

    If you draw an equilateral triangle between the three points of intersection of the smaller and large circles you can work out that

    $r = \dfrac{\sqrt{3}}{2+\sqrt{3}}R$

    see if that helps you work out the rest.
    Last edited by romsek; Mar 14th 2017 at 09:11 AM.
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  3. #3
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    Re: Three equal circle in the big circle

    and how i can calculate blue figure area? it's not circle or triangle and i dont know formule of blue figure area
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  4. #4
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    Re: Three equal circle in the big circle

    Quote Originally Posted by Lortkiphanidze16 View Post
    and how i can calculate blue figure area? it's not circle or triangle and i dont know formule of blue figure area
    Blue area = (Area of equilateral triangle formed by connecting the small circle centers) - (1/2 area of a small circle)
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  5. #5
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    Re: Three equal circle in the big circle

    Quote Originally Posted by skeeter View Post
    Blue area = (Area of equilateral triangle formed by connecting the small circle centers) - (1/2 area of a small circle)
    while Skeeter is a wonder I didn't think his math intuition was supernatural so I figured this must be a known problem and sure enough...


    https://owlcation.com/stem/Solution-...ircles-Problem
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    Re: Three equal circle in the big circle

    Quote Originally Posted by romsek View Post
    while Skeeter is a wonder I didn't think his math intuition was supernatural so I figured this must be a known problem and sure enough...


    https://owlcation.com/stem/Solution-...ircles-Problem
    Not super-anything ... just sketched it

    triangle area - 3 sectors of 60 degrees

    Three equal circle in the big circle-circle_area.png
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    Re: Three equal circle in the big circle

    Quote Originally Posted by skeeter View Post
    Not super-anything ... just sketched it

    triangle area - 3 sectors of 60 degrees

    Click image for larger version. 

Name:	circle_area.png 
Views:	2 
Size:	6.2 KB 
ID:	37187
    well I think that's pretty clever so there!
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  8. #8
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    Re: Three equal circle in the big circle

    Can you tell me from where this formula : r=3–√2+3–√R?

    its depend on the radius or it's general formula?

    Thanks
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    Re: Three equal circle in the big circle

    Quote Originally Posted by Lortkiphanidze16 View Post
    Can you tell me from where this formula : r=3√2+3√R?
    its depend on the radius or it's general formula?
    Three equal circle in the big circle-11973278_f1024.jpg
    The larger radius is $R=x+y$ and the smaller is $r=x$
    Now $\frac{x}{y}= \sin\left(\frac{\pi}{3}\right)= \left(\frac{\sqrt3}{2}\right)$
    Can you put it all together?
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