Question

arc ab is \\(\frac{1}{3}\\) of the circumference of a circle. what is the radian measure of the central angle?
\\(\frac{\pi}{3}\\) radians
\\(\frac{2\pi}{3}\\) radians
\\(\frac{3\pi}{2}\\) radians
\\(3\pi\\) radians

Explanation:

Step1: Recall the total radians in a circle

A full circle has a central angle of \(2\pi\) radians (since the circumference corresponds to a central angle of \(2\pi\) radians).

Step2: Calculate the central angle for arc AB

Given that arc AB is \(\frac{1}{3}\) of the circumference. So the central angle (in radians) corresponding to arc AB is \(\frac{1}{3}\) of \(2\pi\) radians.
Mathematically, that is \(\frac{1}{3}\times2\pi=\frac{2\pi}{3}\) radians.

Answer:

\(\frac{2\pi}{3}\) radians (the option with \(\frac{2\pi}{3}\) radians)