Question

arc cd is $\frac{2}{3}$ of the circumference of a circle. what is the radian measure of the central angle?
$\frac{2pi}{3}$ radians
$\frac{3pi}{4}$ radians
$\frac{4pi}{3}$ radians
$\frac{3pi}{2}$ radians

Explanation:

Step1: Recall the relationship between arc - length, circumference and central angle

The measure of a full - circle in radians is \(2\pi\) radians. If an arc is a fraction of the circumference of a circle, the radian measure of the central angle corresponding to that arc is the same fraction of \(2\pi\).

Step2: Calculate the radian measure of the central angle

The arc CD is \(\frac{2}{3}\) of the circumference of the circle. So the radian measure of the central angle \(\theta\) is \(\theta=\frac{2}{3}\times2\pi=\frac{4\pi}{3}\) radians.

Answer:

C. \(\frac{4\pi}{3}\) radians