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?
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