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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?

 

 Oct 25, 2021
 #1
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Since the circumference is 36, we know that the radius of each circle is 9/pi. Then, we can connect the centers of the circles together in order to get an equaliteral triangle, since the circles are the same. From this, we know that the equaliteral triangle will have a side length of 18/pi. We can then find the 60 degree arcs that the equaliteral triangle creates. Therefore, each 60 degree arc is 1/6 * (9/pi)^2 * pi = 27/2pi. 

 

Now, we can calculate the area of the equaliteral triangle which is simply 81sqrt(3)/pi^2. Then, we can subtract the arcs in order to get the shaded area. Therefore, the region we are looking for is:

 

\(\frac{81\sqrt{3}}{\pi^2} - (3 \cdot \frac{27}{2\pi}) = \frac{162\sqrt{3}-81\pi }{2\pi ^2}\)

 

The way I did it is correct, however some of the numbers may be wrong so please double check before submitting this solution. 

 Oct 25, 2021
 #2
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Radius = 1/2(36/pi) = 18/pi

 

Triangle side (a) = 2(18/pi)

 

Triangle area = a2(√3 / 4)

 

60º sector area = 1/6[(18/pi)2*pi]

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