#### Question

The area of a sector of a circle of 6 cm radius is 15 \[\pi\] sq.cm. Find the measure of the arc and length of the arc corresponding to the sector.

#### Solution

The 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 \[\pi\] cm^{2} (Given)

\[\therefore \frac{1}{2}lr = 15\pi\]

\[ \Rightarrow \frac{1}{2} \times l \times 6 = 15\pi\]

\[ \Rightarrow l = \frac{15\pi}{3} = 5\pi \text{ cm} \]

Length of the arc = \[5\pi \text{ cm} \]

\[\therefore \frac{\theta}{360°} \times 2\pi r = 5\pi\]

\[ \Rightarrow \theta = \frac{5 \times 360°}{2r}\]

\[ \Rightarrow \theta = \frac{5 \times 360° }{2 \times 6}\]

\[ \Rightarrow \theta = 150° \]

Thus, the measure of the arc and length of the arc corresponding to the sector are 150º and 5 \[\pi\] cm, respectively.