Chapter 11.3, Problem 32PPS

To determine

The area of the shaded region.

Answer to Problem 32PPS

The area of the shaded region is 69.7 sq. ft. (approx.).

Explanation of Solution

Given: A circle enclosed within a square of side 18 ft as shown below-\

  

Concept Used: The formula for the area of circle with radius r is given by-

Area of circle $=\pi r^2$

The formula for the area of a square with side a is given by-

Area of square $=a^2$

The area of any shaded region can be found by subtracting the unshaded region from the total region.

Calculations: The total region in this case is a square with side 18 ft. Thus, area of total region-

Total area = Area of the square

  $={(18)}^2$ sq. ft

  $=324$ sq. ft

The unshaded region is a square whose diameter is equal to that of the side of the square as every side is a tangent to the circle. Hence, diameter of the circle is also 18 ft. Hence, the radius of the circle is given by-

Radius $=\frac{18}{2}=9$ ft

Therefore, area of the unshaded region is given by-

Unshaded area = Area of circle

  $=\pi {(9)}^2$ sq. ft.

  $=3.14\times 81$ sq. ft.

  $=254.34$ sq. ft.

Therefore, total area of the shaded region is-

Shaded area = Area of square $-$ Area of circle

  $=324-254.34$ sq. ft.

  $=69.66$ sq. ft.

Thus, area of the shaded region is 69.7 sq. ft (approx.).