Chapter 12.5, Problem 50SR

To determine

To find the area of each shaded region.

Answer to Problem 50SR

  $55$ square centimeters

Explanation of Solution

Given:

  $$

Circle:

Diameter $=20$ inches

Hexagon

Formula used:

Area of a circle $=\pi r^2$ , where $r$ is the radius of the circle.

Area of a hexagon $=\frac{3\sqrt{3}}{2}{a}^2$ , where $a$ is the side of the hexagon.

Calculation:

  

From the figure,

  $AB=$ diameter of the circle$\Rightarrow d=20$

  $\Rightarrow r=\frac{d}{2}$

  $\Rightarrow r=10$ inches

Now for the area of circle,

  $\Rightarrow \pi r^2$

Putting the value of $r=10,$

  $\Rightarrow 3.14\times 10\times 10$

  $\Rightarrow 314$ square inches.

So, area of the circle $=314$ square inches.

Now for the area of hexagon,

Radius of the circle $=$ side of the hexagon $=$

  $a$ = $10$ inches.

Area of the hexagon $=\frac{3\sqrt{3}}{2}{a}^2$

Not putting the value of $a=10$ ,

  $\Rightarrow \frac{3\sqrt{3}}{2}{(10)}^2$

  $\Rightarrow 259.81$ square inches

So, area of the shaded region $=$ Area of the circle $-$ Area of the hexagon

  $\Rightarrow 314-259.81$

  $\Rightarrow 54.91$ square inches

Hence, area of the shaded region $\approx 55$ square centimeters.

Conclusion:

Therefore, the area of the shaded region in the given figure is $55$ square centimeters.