Question

Find the area of a shaded region in the given figure, where a circular arc of radius $7$ cm has been drawn with vertex A of an equilateral triangle ABC of side $14$ cm as centre. (Use $\pi$ $=\frac{22}{7}$ and $\sqrt{3}$ $=1.73$).

Answer 1

$\Rightarrow$ Here, Radius of circle $r=7cm$

$\Rightarrow$ Area of circle $=\pi r^2=\frac{22}{7}\times (7{)}^2=154c{m}^2$

$\Rightarrow$ Side of equilateral triangle is $14cm$.

$\Rightarrow$ Area of equilateral triangle $=\frac{\sqrt{3}}{4}\times (side{)}^2$

$\Rightarrow$ Area of equilateral triangle $=\frac{\sqrt{3}}{4}\times (14{)}^2$

$\Rightarrow$ Area of equilateral triangle $=\frac{1.73}{4}\times 196=84.87c{m}^2$

$\Rightarrow$ $\angle A={60}^o$ [ Angle of equilateral triangle ]

$\therefore$ $\theta ={60}^o$

$\Rightarrow$ Area of sector $=\frac{\theta }{{360}^o}\times \pi r^2$

$\Rightarrow$ Area of sector $=\frac{{60}^o}{{360}^o}\times \frac{22}{7}\times (7{)}^2$

$\therefore$ Area of sector $=25.66c{m}^2$

$\Rightarrow$ Area of shaded region $=$ (Area of circle + Area of equilateral triangle ) - $2\times$Area of sector

$\Rightarrow$ Area of shaded region $=(154+84.87)-51.32=187.55c{m}^2$