Multiple choice

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.

  1. $\pi - 2$
  2. $\pi + 2$
  3. $\dfrac {\pi}{3}$
  4. $\dfrac {2\pi}{3}$
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A Correct answer
Explanation

The perimeter of a sector is 2r + r*theta, where r is radius and theta is the central angle. The length of a semicircle arc is pi*r. Setting 2r + r*theta = pi*r, we divide by r to get 2 + theta = pi, so theta = pi - 2.

AI explanation

Let the radius of the sector and the semicircle be r. The length of the arc of the semicircle is pi*r, and the perimeter of the sector is r + r + L, where L is its arc length. Equating the two gives 2r + L = pi*r, so the arc length of the sector is L = r*(pi - 2). Using the arc length formula L = r*theta, we find the angle at the centre is pi - 2.

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