question_answer

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 area of the shaded region.

Answer:

Given, radius of large semi-circle \[=4.5\text{ }cm\]

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 \[=4.5\text{ }cm\]

\[\Rightarrow \]                           \[r=\frac{4.5}{2}cm\]

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 \[=3\text{ }cm\]

\[r=\frac{3}{2}cm\]

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

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

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

\[=\frac{11}{7}\times \frac{90}{4}-\frac{22}{7}\times \frac{29.25}{4}\]

\[=\frac{990-643.5}{28}=\frac{346.5}{28}\]

\[=12.37\,\,c{{m}^{2}}\,\](approx).