Question : An arc of length $33 \pi\;\mathrm{cm}$ subtends an angle of $132^\circ$ at the centre of the circle. Find the radius of the circle.
Option 1: $45\;\mathrm{cm}$
Option 2: $40\;\mathrm{cm}$
Option 3: $30\;\mathrm{cm}$
Option 4: $35\;\mathrm{cm}$
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Correct Answer: $45\;\mathrm{cm}$
Solution :
The length of an arc $(L)$ in a circle,
$L = 2 \pi r \left(\frac{\theta}{360^\circ}\right)$
where $r$ is the radius of the circle and $θ$ is the angle subtended by the arc at the centre of the circle in degrees.
Given that $L = 33 \pi\;\mathrm{cm}$ and $θ = 132^\circ$,
So, $33\pi = 2 \pi r \left(\frac{132^\circ}{360^\circ}\right)$
$⇒r = 33 \times \frac{360^\circ}{132^\circ \times 2} = 45 \text{ cm}$
Hence, the correct answer is $45 \text{ cm}$.
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