Example:
In the above diagram, find the area of the sector \(OAB\).
Based on the question, area of the sector \(OAB\)
\(\begin{aligned} &=\dfrac{1}{2}r^2 \theta \\\\ &=\dfrac{1}{2}(9)^2(1.3) \\\\ &= 52.65 \text{ cm}^2. \end{aligned}\)
The diagram shows a sector of a circle, with centre \(O\) and a radius of \(7 \text{ cm}\).
The length of the arc \(AB\) is \(5 \text{ cm}\).
Find the area of the shaded region.
Length of the arc, \(s = r\theta = AB \)
Then,
\(\begin{aligned} r\theta&=5 \\\\ 7\theta &=5 \\\\ \theta&=\dfrac{5}{7} \text{ rad}.\\\\ \angle AOB&=\dfrac{5}{7}\times \dfrac{180}{3.\,142} \\\\ &=40.92^\circ \end{aligned}\)
Area of the shaded region
\(\begin{aligned} &=\dfrac{1}{2}r^2(\theta-\sin \theta) \\\\ &=\dfrac{1}{2}(7)^2 \begin{pmatrix} \dfrac{5}{7}-\sin40.92^\circ \end{pmatrix} \\\\ &=\dfrac{1}{2}(49)(0.\,0593) \\\\ &=1.\, 4529 \text{ cm}^2.\end{aligned}\)
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