An arc of length 3 miles lies on a circle of radius 5 miles. Find the central angle it subtends.

Step-by-step solution

1. 1

   Rearrange the arc-length formula. From $s=r\theta$,

   $\theta=\frac{s}{r}$

   This single ratio is what radian measure is — arc divided by radius — which is why the answer will carry no physical unit.

2. 2

   Substitute.

   $\theta=\frac{3}{5}=0.6$

3. 3

   State the answer with its (lack of) units. The miles cancel, so

   $\boxed{\theta=0.600\ \text{radians}}$

4. 4

   Check by going forwards. $s=r\theta=5(0.6)=3$ miles ✓.

5. 5

   Convert to degrees for intuition. $0.6\times\dfrac{180}{\pi}=34.38^{\circ}$. Since the arc ($3$ mi) is shorter than the radius ($5$ mi), the angle must be less than $1$ radian $\approx 57.3^{\circ}$ — and it is.