Question

Three semicircles each of diameter 3 cm, a circle of diameter 4.5 cm and a semicircle of radius 4.5 cm are drawn in the given figure. Find the are of the shaded region.

Answer 1

Given, radius of large semi-circle(R) = 4.5 cm

Area of large semi-circle = $\frac{1}{2}\pi R^2$

= $\frac{1}{2}\times \frac{22}{7}\times 4.5\times 4.5$

Diameter of inner circle = 4.5 cm

$\Rightarrow r=\frac{4.5cm}{2}$

Area of inner circle = $\pi r^2$

= $\frac{22}{7}\times \frac{4.5}{2}\times \frac{4.5}{2}$

Diameter of small semi-circle = 3cm
r = $\frac{3}{2}$ cm

Area of small semi-circle = $\frac{1}{2}\pi r^2$

=$\frac{1}{2}\times \frac{22}{7}\times \frac{3}{2}\times \frac{3}{2}$

Area of shaded region = Area of large semi circle + Area of 1 small semi circle - Area of inner circle - Area of 2 small semi circle

= $\frac{1}{2}\times \frac{22}{7}\times 4.5\times 4.5+\frac{1}{2}\times \frac{22}{7}\times \frac{3}{2}\times \frac{3}{2}-\frac{22}{7}\times \frac{4.5}{2}\times \frac{4.5}{2}-2\times \frac{1}{2}\times \frac{22}{7}\times \frac{3}{2}\times \frac{3}{2}$

=$\frac{1}{2}\times [20.25+\frac{9}{4}]-\frac{22}{7}[\frac{20.25}{4}+\frac{9}{4}]$

=$\frac{1}{2}\times \frac{90}{4}-\frac{22}{7}\times \frac{29.25}{4}$

= $\frac{990-643.5}{25}=\frac{346.5}{28}$

=$12.37c{m}^2(approx.)$