Chapter 11.3, Problem 37PPS

To determine

The area of the shaded region.

Answer to Problem 37PPS

The area of the shaded region is 7.9 $i{n}^2$ .

Explanation of Solution

Given: Three shaded sectors inside a circle with radius 2 in as shown below-

  

Concept Used: The formula for the area of the sector of a circle with central angle $\theta$ and radius r is given by-

Area of sector $=\frac{\theta }{360°}\times \pi r^2$

Total angle at the centre of any circle is $360°$ .

Calculations: The unshaded region consists of three congruent sectors of the circle with central angle $45°$ . Thus, total central angle included in the unshaded region

  $=3\times 45°$

  $=135°$

Hence, total central angle included within the shaded region is

  $=360°-135°$

  $=225°$

The shaded region consists of three congruent sectors each with radius 2 in. Let the central angle of the shaded sectors be x. Thus,

  $x+x+x=225°$

  $\Rightarrow x=75°$

Hence, the central angle of each shaded sector is $75°$ . Thus, total area of the shaded region is given by-

Shaded area $=$ $3\times$ (Area of one shaded sector)

  $=3\times \frac{75°}{360°}\times \pi {\left(2\right)}^2$ $i{n}^2$

  $=\frac{225°}{360°}\times 3.14\times 4$ $i{n}^2$

  $=\frac{5}{8}\times 3.14\times 4$ $i{n}^2$

  $=7.9$ $m^2$ (approx.)

Thus, area of the shaded region is $7.9$ $i{n}^2$ .