EXAMPLE 2.6.4

The diagram below shows a circular pizza whose diameter is 8 inches, situated in a square box whose side length is 10 inches. How much of the box is "empty?"

SOLUTION

The area of the "empty" part of the box is the area of the shaded region of the figure shown above. Since the shaded region corresponds to the region the remains after a circle has been removed from a square, we make this observation about area:

Area of shaded region = area of square - area of circle.

The area of the square is computed as follows:

Area of rectangle = LW = (10 in.)(10 in.) = 100 sq. in.

To compute the area of the circle, we first observe that since the diameter of the circle is 8 inches, its radius is 4 inches. Now we use the formula for the area of a circle:

Now we subtract:

Area of shaded region = 100 - 50.24 = 49.76 sq. in.

Comparing the area of the "empty part" of the box (49.76 sq. in.) with the area of the entire box (100 sq. in.), we see that when a 10 inch by 10 inch square box is used to contain an 8-inch-diameter circular pizza, the box is approximately half empty.