Question : In the following figure, the length of arc AB is equal to five times the radius $r$ of the circle. Find the area of sector AOB.
Option 1: $2.5 r^2$
Option 2: $\frac{1}{2}r^2$
Option 3: $3r^2$
Option 4: $2r^2$
Correct Answer: $2.5 r^2$
Solution :
Length of the arc AB = $5r$
Radius of the circle = $r$
Given:
Length of arc = $5r$
⇒ $\frac{\theta}{360^\circ }\times 2\pi r=5r$
⇒ $\theta = \frac{360^\circ}{2\pi} \times 5$
Area of sector with angle $\theta$ = $\frac{\theta}{360^\circ}\pi r^2$
=$\frac{\frac{360^\circ}{2\pi} \times 5}{360^\circ}\pi r^2$
= $2.5r^2$
Hence, the correct answer is $2.5r^2$.
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