Question

Find the area of the shaded region.

Answer 1

Hint: To find the area of the shaded region in the given figure we will subtract the area of the circle having radius 1 cm and the two semi-circle having radius 2 cm from the bigger semi-circle of radius 4 cm. Then we will get the area of the shaded region.

Complete step-by-step answer:
We have been given the figure as follows:

From the given figure, we can say that the area of the shaded region is equal to the area obtained by subtracting the area of the circle with C as center and the two semi-circles with AM and MB as diameter from the bigger semi-circle having AB as diameter.
We know that area of a circle having D as diameter is as follows:
Area $$={\displaystyle \frac{\pi D^2}{4}}$$
So, area of semi-circle $$={\displaystyle \frac{1}{2}}\times$$ area of circle
$$\begin{array}{rl}& ={\displaystyle \frac{1}{2}}\times {\displaystyle \frac{\pi D^2}{4}}\\ & ={\displaystyle \frac{\pi D^2}{8}}\end{array}$$
Now the area of semi-circle having AM = 4 cm as diameter $$={\displaystyle \frac{\pi D^2}{8}}$$
$$={\displaystyle \frac{\pi \times {\left(4\right)}^2}{8}}={\displaystyle \frac{\pi \times 16}{8}}=2\pi c{m}^2$$
Area of semi-circle having MB = 4 cm as diameter $$={\displaystyle \frac{\pi D^2}{8}}$$
$$={\displaystyle \frac{\pi \times 4}{8}}={\displaystyle \frac{\pi \times 16}{8}}=2\pi c{m}^2$$
Area of circle having C as center and diameter $$=(2\times 1)cm$$ is $$={\displaystyle \frac{\pi D^2}{4}}$$
$$={\displaystyle \frac{\pi \times {\left(2\right)}^2}{4}}=\pi c{m}^2$$
Area of semicircle having AB = 8 cm as diameter $$={\displaystyle \frac{\pi D^2}{8}}$$
$$={\displaystyle \frac{\pi \times 8^2}{8}}=\pi \times 8=8\pi c{m}^2$$
So the area of the shaded region = area of semi-circle having AB as diameter – (area of the circle having C as center + 2 area of semi-circle with AM as diameter)
$$=8\pi -(\pi +2\times 2\pi )=8\pi -5\pi =3\pi c{m}^2$$
Also, $$\pi ={\displaystyle \frac{22}{7}}$$ so by substituting the value of $$\pi$$ we get as follows:
Area of shaded region $$=3\pi =3\times {\displaystyle \frac{22}{7}}=9.43c{m}^2(approx)$$
Therefore, the area of the shaded region is $$9.43c{m}^2(approx)$$.
Note: Be careful while calculating the area as we have been given the diameter of some circle and radius of a circle so accordingly use the formula to calculate the area.