# Thread: Three equal circle in the big circle

1. ## 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

2. ## 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.

3. ## 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

4. ## Re: Three equal circle in the big circle

Originally Posted by Lortkiphanidze16
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)

5. ## Re: Three equal circle in the big circle

Originally Posted by skeeter
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

6. ## Re: Three equal circle in the big circle

Originally Posted by romsek
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

7. ## Re: Three equal circle in the big circle

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

triangle area - 3 sectors of 60 degrees

well I think that's pretty clever so there!

8. ## 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

9. ## Re: Three equal circle in the big circle

Originally Posted by Lortkiphanidze16
Can you tell me from where this formula : r=3–√2+3–√R?
its depend on the radius or it's general formula?

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?