What is the length of the arc on the unit circle subtended by an angle of $120^\circ$? Show all work.

Question

What is the length of the arc on the unit circle subtended by an angle of $120^\circ$? Show all work.

1. $\dfrac{2}{3}$
2. $\dfrac{1}{3}\pi$
3. $\dfrac{2}{3}\pi$
4. $\pi$

I used an equation where the central angle equals the arc length divided by the radius. Since $120^\circ$ degrees is $\frac 13$ of the total circumference of $360^\circ$ (or $2\pi$), I chose the answer to be $2$.

Is my answer correct? If not, please explain why.

Answer 1

Angle, $ \theta\ (\text{in radians}) = \dfrac{\text{arc-length}}{\text{radius}} = \dfrac{s}{r}$

We have $r=1$ and $\theta = 120^\circ = \dfrac{2\pi}{3}$

So, $s=r\theta = 1\cdot \dfrac{2\pi}{3} = \dfrac{2\pi}{3}$

Answer 2

The unit circle has perimeter $2\pi$. $120^\circ$ is $\frac{120^\circ}{360^\circ}= \frac 13$ of the entire circle $\frac13\cdot2\pi$ is $\frac{2\pi}{3}$.