Question

arc ab is 1/6 of the circumference of a circle. what is the radian measure of the central angle?

Explanation:

Step1: Recall 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 radian measure of the central angle. Also, the circumference of a circle is $C = 2\pi r$. Given that the arc length $s$ is $\frac{1}{6}$ of the circumference, so $s=\frac{1}{6}\times2\pi r=\frac{\pi r}{3}$.

Step2: Substitute into arc - length formula

Since $s = r\theta$ and $s=\frac{\pi r}{3}$, we substitute $s$ into the arc - length formula: $\frac{\pi r}{3}=r\theta$.

Step3: Solve for $\theta$

Divide both sides of the equation $\frac{\pi r}{3}=r\theta$ by $r$ (assuming $r
eq0$). We get $\theta=\frac{\pi}{3}$.

Answer:

$\frac{\pi}{3}$