Question

find the area ( a ) of the sector and the arc length ( s ) of a circle of radius 5 feet formed by the central angle of ( 60^circ )

Explanation:

Step1: Convert degrees to radians

To use sector and arc length formulas, convert $60^\circ$ to radians:
$$60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3}$$

Step2: Calculate sector area $A$

Use the sector area formula $A=\frac{1}{2}r^2\theta$, where $r=5$, $\theta=\frac{\pi}{3}$:
$$A=\frac{1}{2} \times 5^2 \times \frac{\pi}{3} = \frac{25\pi}{6}$$

Step3: Calculate arc length $s$

Use the arc length formula $s=r\theta$, where $r=5$, $\theta=\frac{\pi}{3}$:
$$s=5 \times \frac{\pi}{3} = \frac{5\pi}{3}$$

Answer:

Sector area: $\frac{25\pi}{6}$ square feet, Arc length: $\frac{5\pi}{3}$ feet