Question

arc ab is \\(\frac{1}{6}\\) of the circumference of a circle. what is the radian measure of the central angle?\
\\(\frac{\pi}{6}\\)\
\\(\frac{\pi}{3}\\)\
\\(\frac{2\pi}{3}\\)\
\\(\frac{5\pi}{6}\\)

Explanation:

Step1: Recall the total radian measure of a circle

A full circle has a central angle of \(2\pi\) radians (since the circumference corresponds to a full rotation, and the radian measure of a full circle is \(2\pi\)).

Step2: Calculate the central angle for arc AB

Since arc AB is \(\frac{1}{6}\) of the circumference, the central angle \(\theta\) corresponding to arc AB is \(\frac{1}{6}\) of the total central angle of the circle. So we calculate \(\theta=\frac{1}{6}\times2\pi\).
Simplifying \(\frac{1}{6}\times2\pi=\frac{2\pi}{6}=\frac{\pi}{3}\)? Wait, no, wait: \(\frac{1}{6}\times2\pi=\frac{2\pi}{6}=\frac{\pi}{3}\)? Wait, no, let's recalculate: \(\frac{1}{6}\times2\pi=\frac{2\pi}{6}=\frac{\pi}{3}\)? Wait, no, 2 divided by 6 is 1/3, so \(\frac{1}{6}\times2\pi = \frac{\pi}{3}\)? Wait, no, wait the total circumference is \(2\pi r\), and the arc length is \(\frac{1}{6}\times2\pi r\). The formula for arc length \(s = r\theta\) (where \(\theta\) is in radians). So \(s=\frac{1}{6}\times2\pi r=r\theta\). Dividing both sides by \(r\) (assuming \(r
eq0\)), we get \(\theta=\frac{2\pi}{6}=\frac{\pi}{3}\)? Wait, no, 2π is the total radians for a full circle. So the fraction of the circle that arc AB is, is the same as the fraction of the total radians (2π) that the central angle is. So if arc AB is \(\frac{1}{6}\) of the circumference, then the central angle is \(\frac{1}{6}\) of \(2\pi\) radians. So \(\theta=\frac{1}{6}\times2\pi=\frac{2\pi}{6}=\frac{\pi}{3}\)? Wait, no, 2π divided by 6 is π/3? Wait, 2π is 360 degrees, so 1/6 of 360 is 60 degrees, and 60 degrees in radians is π/3 (since π radians is 180 degrees, so 60 degrees is π/3 radians). Wait, but let's check the options. The options are π/6, π/3, 2π/3, 5π/6. Wait, maybe I made a mistake. Wait, no: the total central angle for a circle is \(2\pi\) radians. So if the arc is 1/6 of the circumference, then the central angle is 1/6 of \(2\pi\). So \(\frac{1}{6}\times2\pi=\frac{\pi}{3}\). So the answer should be π/3. Wait, but let's re-express: \(2\pi\) radians is 360 degrees. 1/6 of 360 is 60 degrees. 60 degrees is \(\frac{\pi}{3}\) radians (since 180 degrees is \(\pi\) radians, so 60 degrees is \(\pi/3\)). So the correct answer is \(\frac{\pi}{3}\).

Answer:

\(\frac{\pi}{3}\) (corresponding to the option \(\frac{\pi}{3}\))