Question

In the following figure, shows a sector of a circle, centre O, containing an angle 0°. Prove that:

(i) Perimeter of the shaded region is $\left(\tan \theta +sec\theta +\frac{\mathrm{\pi \theta }}{180}-1\right)$

(ii) Area of the shaded region is $\frac{r^2}{2}\left(\tan \theta -\frac{\mathrm{\pi \theta }}{180}\right)$

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Answer 1

It is given that the radius of circle is r and the angle.

In,

It is given that.

(i) We know that the arc length l of a sector of an angle θ in a circle of radius r is

Now we substitute the value of OC, OA and l to find the perimeter of sector AOC,

Hence,

(ii) We know that area A of the sector at an angle θ in the circle of radius r is

.

Thus

Hence,