In the figure above, three tangent circles with their centers along the same horizontal axis are shown. The biggest circle contains two smaller circles. The larger of the two smaller circles has a radius of 6 inches and the other has a radius of 3 inches. What is the area of the shaded region (in square inches)?

A. 4π
B. 9π
C. 18π
D. 36π
E. 72π

(C) Tangent circles are circles that touch each other, but do not overlap, nor do they leave any gaps between them. If the larger circle has a radius of 6 and the small one a radius of 3, we can calculate the diameter of each as 12 and 6. The diameter of the largest circle, then, would be 18. To find the radius, divide the diameter by 2: 18/2 = 9.

In order to calculate the area of the shaded region, you first need to calculate the area of the largest circle. Using the formula A = πr², we find that the area of the largest circle is 81π. Now subtract from it that area of the two smaller circles, each of which is 9π and 36π, respectively. 81π – 36π – 9π = 36π.

Because the circles are tangential and their centers all lie on the same line, all we have to do now to find the area of the shaded region is to divide the result by 2: 36π/2 = 18π.

The answer is (C).
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Why is the area of the shaded region not the Area of the large circle minus the areas of the 2 smaller circles? Why do we divide by 2?

The area of the largest circle minus the areas of the 2 smaller circles is the shaded area on the following picture:

We can see that it is twice the requested area.

Another way to look at this problem is to do same calculations for semi-circles: