Find the length of the arc of a circle of radius 3 metres subtended by a central angle of 120 degrees, rounded to three decimal places.

Step-by-step solution

1. 1

   Convert the angle to radians. The formula $s=r\theta$ holds only when $\theta$ is in radians:

   $\theta=120^{\circ}\times\frac{\pi}{180}=\frac{2\pi}{3}$

2. 2

   Apply the arc-length formula.

   $s=r\theta=3\times\frac{2\pi}{3}$

3. 3

   Simplify. The $3$ cancels:

   $s=2\pi$

4. 4

   Convert to a decimal.

   $s=2\pi=6.28319\approx 6.283\ \text{m}$

   $\boxed{s=2\pi\approx 6.283\ \text{m}}$

5. 5

   Check as a fraction of the circumference. The full circumference is $2\pi r=6\pi$, and $120^{\circ}$ is one third of a full turn, so the arc should be $\tfrac13(6\pi)=2\pi$ ✓ — matching the formula and confirming the degree-to-radian conversion.