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Shaded Regions
 

To find the area of the shaded region of a figure, subtract the area of the unshaded region from the area of the entire figure.
 

Example 1:

What is the area of the shaded region formed by the circle and the rectangle in the figure to the right?



(A) 15 –– 2π
(B) 15 ––π
(C) 14
(D) 16 ––π
(E) 15π

 
To find the area of the shaded region subtract the area of the circle from the area of the rectangle:

area of rectangle - area of a circle
3·5 - π • 12
15 - π

The answer is (B).

 

Example 2:

In the figure below, the radius of the larger circle is three times that of the smaller circle. If the circles are concentric, what is the ratio of the shaded region’’s area to the area of the smaller circle?



(A) 10:1
(B) 9:1
(C) 8:1
(D) 3:1
(E) 5:2

 

Since we are not given the radii of the circles, we can choose any two positive numbers such that one is three times the other. Let the outer radius be 3 and the inner radius be 1. Then the area of the outer circle is π32 = 9π , and the area of the inner circle is π12 = π . So the area of the shaded region is 9π –– π = 8π. Hence, the ratio of the area of the shaded region to the area of the smaller circle is 8ππ = 81. Therefore, the answer is (C).





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