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)

$\pi -2$

Answer 2

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.
$\therefore\,\,\,\, \frac{\theta}{360^{\circ}} \times 2 \pi r+2 r=\pi r$
$\Rightarrow\,\,\,\,\, \theta+2=\pi$
$\Rightarrow\,\,\,\,\, \theta=\pi-2$