Based on the diagram above, area of sector can be write as:
\(\text{Area}=\dfrac{1}{2}r^2\theta\)
where \(r\) is the radius and \(\theta\) is the angle in radians.
Another formula to calculate the area of a sector is given by:
\(\text{Area}=\dfrac{\theta}{360^\circ}\times\pi r^2\)
where \(r\) is the radius and \(\theta\) is the angle in degrees.
Based on figure above, area of segment can be calculate by:
\(= \text{Area of the sector } OPRQ - \text{Area of the isosceles }\triangle{OPQ} \ \)
or can be calculated by using formula:
where \(j\) is the radius of the sector and \(\theta\) is the angle of the sector.
Based on the diagram above, find the area of sector \(AOB\).
Based on the figure, given,
\(r=9\) cm, \(\theta=1.3\text{ rad}\).
Area of sector \(AOB\):
\(\begin{aligned} &=\dfrac{1}{2}r^2 \theta \\\\ &=\dfrac{1}{2}(9)^2(1.3) \\\\ &= 52.65 \text{ cm}^2. \end{aligned}\)
Diagram above shows a sector of a circle, with centre \(O\) and a radius of \(7\) cm.
The length of the arc \(AB\) is \(5\) cm.
Find the area of the shaded region.
[ Use \(\pi=3.142\) ]
Find the value of \(\theta\) by using arc length formula,
\(\begin{aligned} s&=r\theta \\\\r\theta&=5 \\\\ 7\theta &=5 \\\\ \theta&=\dfrac{5}{7} \text{ rad}\\\\ \angle AOB&=\dfrac{5}{7}\times \dfrac{180}{3.\,142} \\\\ &=40.92^\circ. \end{aligned}\)
Area of 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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