Question

in circle o, central angle aob measures \\(\frac{\pi}{3}\\) radians. what is the length of arc ab? \\(\bigcirc\\) \\(6\pi\\) cm \\(\bigcirc\\) \\(12\pi\\) cm \\(\bigcirc\\) \\(18\pi\\) cm \\(\bigcirc\\) \\(36\pi\\) cm (with an image of circle o, points a, b on the circle, oa and ob as radii labeled 18 cm)

Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( s \) when the central angle \( \theta \) is in radians is \( s = r\theta \), where \( r \) is the radius of the circle.

Step2: Identify radius and angle

From the diagram, the radius \( r = 18 \) cm and the central angle \( \theta=\frac{\pi}{3} \) radians.

Step3: Calculate arc length

Substitute \( r = 18 \) and \( \theta=\frac{\pi}{3} \) into the formula: \( s=18\times\frac{\pi}{3} \). Simplify the expression: \( 18\times\frac{\pi}{3}=6\pi \) cm.

Answer:

\( 6\pi \) cm (corresponding to the option "6π cm")