Question

The area of a sector of a circle of 6 cm radius is 15$\mathrm{\pi }$ sq.cm. Find the measure of the arc and length of the arc corresponding to the sector.

Answer 1

Radius of the sector, r = 6 cm

Let the measure of the arc be θ and the length of the arc corresponding to the sector be l cm.

Area of the sector = 15$\mathrm{\pi }$ cm² (Given)

$$\begin{gathered} \therefore \frac{1}{2}lr=15\mathrm{\pi } \\ \Rightarrow \frac{1}{2}\times l\times 6=15\mathrm{\pi } \\ \Rightarrow l=\frac{15\mathrm{\pi }}{3}=5\mathrm{\pi }\mathrm{cm} \end{gathered}$$
Length of the arc = $5\mathrm{\pi }\mathrm{cm}$

$$\begin{gathered} \therefore \frac{\theta }{360°}\times 2\mathrm{\pi }r=5\mathrm{\pi } \\ \Rightarrow \theta =\frac{5\times 360°}{2r} \\ \Rightarrow \theta =\frac{5\times 360°}{2\times 6} \\ \Rightarrow \theta =150° \end{gathered}$$
Thus, the measure of the arc and length of the arc corresponding to the sector are 150º and 5$\mathrm{\pi }$ cm, respectively.