Paper 1 · circle geometry / radians

In a circle of radius $r$ and centre $O$ , the shaded region is bounded by the chord $AC$ , the diameter $AB$ and the arc $BC$ , with $\widehat{BAC} = \theta$ radians. Find an exact expression for the perimeter of the shaded region.

2r(1 + \theta + \cos\theta)

r(1 + 2\theta + \cos\theta)

2r(1 + \theta + \sin\theta)

2r(\theta + \cos\theta)

Explanation:

AB

is a diameter so

\widehat{ACB}=90°

, giving

AC=2r\cos\theta

. Arc

BC

subtends a central angle

2\theta

, so its length is

2r\theta

. The perimeter is

AC+\text{arc }BC+AB=2r\cos\theta+2r\theta+2r

Derived from ZIMSEC Mathematics Paper 1, November 2011, Q7

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