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# Six small circles, each of radius 3 units, are tangent to a large circle as shown.

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Six small circles, each of radius 3 units, are tangent to a large circle as shown. Each small circle also is tangent to its two neighboring small circles.  What is the area of the region that is inside the large circle but outside the small circles? Nov 12, 2020

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Well....of course......the combined area of the  six small circles is just  6 * pi (radius)^2  = 6 pi (3)^2  = 54pi

Note that  if   we  connect  the centers  of  the top two circles  and we  connect the center of the  large circle with both of these centers,, we  will  have an isosceles triangle  with one side of 6 , two sides of x  and an included angle  between these unknown sides of 60°

Using the Law of Cosines, we can find  x  thusly......

6^2  = 2x^2  -  2x^2  (cos 60)

36  = 2x^2 - 2x^2 (1/2)

36  = x^2

x  = 6

So.....the radius of the larger circle  = x + 3  =   6 + 3   =   9

So  the area  inside rhe larger circle  but  outside the smaller circles =

pi  [ 9^2  - 54]  =

pi [ 81 - 54 ]  =

27 pi  units^2   Nov 12, 2020