Question 4.1.8: Finding the Length of a Circular Arc A circle has a radius o...

The formula s = rθ can be used only when θ is expressed in radians. Thus, we begin by converting 120° to radians. Multiply by \frac{\pi \text { radians }}{180^{\circ}}.

120^{\circ}=120^{\cancel{\circ}} \cdot \frac{\pi \text { radians }}{180^{\cancel{\circ}}}=\frac{120 \pi}{180} \text { radians }=\frac{2 \pi}{3} \text { radians }

Now we can use the formula s = rθ to find the length of the arc. The circle’s radius is 10 inches: r = 10 inches. The measure of the central angle, in radians, is \frac{2 \pi}{3}: \theta=\frac{2 \pi}{3}. The length of the arc intercepted by this central angle is

s=r \theta=(10 \text { inches })\left(\frac{2 \pi}{3}\right)=\frac{20 \pi}{3} \text { inches } \approx 20.94 \text { inches. }