Question

The perimeter of a certain sector of a circle is equal to the length of the arc of the semicircle. Then the angle at the centre of the sector in radians is

• A. $\pi -2$
• B. $\pi +2$
• C. $\frac{\pi }{3}$
• D. $\frac{2\pi }{3}$

Answer 1

Answer: (A)

Let the radius of circle be $r$.
$\therefore$ Length of an arc $=\frac{\theta }{360^{\circ }}\times 2\pi r$
Since, perimeter of a sector of a circle $=$ length of the arc of the semicircle.
$\therefore \frac{\theta }{360^{\circ }}\times 2\pi r+2r=\pi r$
$\Rightarrow \theta +2=\pi$
$\Rightarrow \theta =\pi -2$