Question

Fig. 15.17, shows a sector of a circle, centre O, containing an angle ${\theta }^{\circ }$. Prove that area of the shaded region is $\frac{r^2}{2}(tan\theta -\frac{\pi \theta }{180})$

Answer 1

Area of shaded region

$=ar△OAB-$area of sector

In $△OAB,\tan \theta =\frac{AB}{OA}=\frac{AB}{r}$

$\Rightarrow AB=r\tan \theta$

Now, required area$=\frac{1}{2}\times OA\times OB-\frac{\theta }{360°}\times \pi r^2$

$=\frac{1}{2}\times r\times r\tan \theta -\frac{\theta }{360°}\pi r^2$

$=\frac{r^2}{2}[\tan \theta -\frac{\pi \theta }{180°}]$