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In the diagram, each of the three identical circles touch the other two. The circumference of each circle is 36. What is the area of the shaded region?

 

 Apr 21, 2021
 #1
avatar+592 
+2

https://gyazo.com/0bb2e632441b774f3d412d086fa55957

 

I made an equilateral triangle using the centers of the triangle. 
Since an equilateral triangle has angles of 60o, each sector part is just 16 of the circle.

 

Circumference of each circle = 36 units 

Radius = 18π

 

Sector areas = 316(18π)2π

12324π2π

12324π

162π

 

Now the area of the equilateral triangle is: 

34(36π)2

341296π2

129634π2

 

Which means the area of the shaded region is: 

129634π2162π

129634π2162ππ2

129634π2648π4π2

12963648π4π2

4(3243162π)4π2

3243162ππ2

 

please tell me if I did anything wrong...

 Apr 21, 2021
 #2
avatar+592 
0

LOL

 

I got something complicated as well... I thought I got it incorrect. i actually got the same answer as you, so nothing's wrong... I think.

 Apr 21, 2021
 #3
avatar+592 
0

lol I was debating whether or not to post that because there was a similar question, however this was asking for area of the shaded region instead of the perimeter of it... cheeky

Logarhythm  Apr 21, 2021
 #4
avatar+37165 
+2

See image below:

Triangle area - cicle sector area * 3 = area in question        

 

sqrt(3)/4 ( 36/ pi)2   - 3  * 1/6 *  pi ( 18/pi)2    = shaded                (we use 1/6 because the sector of the circle is 60/360 of the circle)

 

324 sqrt(3) / pi2    - 162 / pi     <============   Same answer as  LR    Way to go Logarhythm!!

 

 

 

 Apr 21, 2021

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