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

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 $\frac{8\mathrm{\pi }}{\mathrm{\pi }}$ = $\frac{8}{1}$. Therefore, the answer is (C).