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Question

The length of an arc of a circle, subtending an angle of $$54^{\circ}$$ at the center is $$16.5\ \text{cm}$$. Calculate the radius, circumference and area of the circle.


Solution

Given:
Length of the arc $$= 16.5 \; \text{cm}$$
$$\theta = 54°$$
To find:
Radius $$= ?$$
Cicumference $$= ?$$
Area of the circle $$=?$$

Length of arc $$= \dfrac{ \theta}{360°}\times 2\pi r$$
$$\Rightarrow \cfrac{2 \times \cfrac{22}{7} \times r \times 54°}{360°} = 16.5$$
$$\Rightarrow r = \cfrac{16.5 \times 20 \times 7}{3 \times 22 \times 2}$$
$$\Rightarrow r = 17.5 \; \text{cm}$$

Now, 
$$\text{Circumference}= 2 \pi r \ = 2 \times \dfrac{22}{7} \times 17.5 \ = 110  \text{ cm}$$
Also,
Area of circle 
$$= \pi {r}^{2} \ = \cfrac{22}{7} \times {\left( 17.5 \right)}^{2} \\ = 22 \times 2.5 \times 17.5 = 962.5 \; \text{cm}^{2}$$

Mathematics
RS Agarwal
Standard X

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