Question

a circle has a radius of 12.6 cm.
what is the exact length of an arc formed by a central angle measuring 120°?
enter your answer in the box. express your answer using π.
cm

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) of a circle is \(s = r\theta\), where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians. First, convert the angle from degrees to radians. We know that \(\theta\) (in radians) \(=\frac{\pi}{180}\times\theta\) (in degrees). Given \(\theta = 120^{\circ}\), then \(\theta=\frac{\pi}{180}\times120=\frac{2\pi}{3}\) radians. The radius \(r = 12.6\) cm.

Step2: Substitute values into the arc - length formula

Substitute \(r = 12.6\) and \(\theta=\frac{2\pi}{3}\) into \(s=r\theta\). So \(s=12.6\times\frac{2\pi}{3}\).

Step3: Simplify the expression

\(12.6\div3 = 4.2\), then \(4.2\times2\pi=8.4\pi\).

Answer:

\(8.4\pi\)