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).
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