An arc of length 2 feet subtends a central angle of 1/3 radian. Find the radius of the circle.

Step-by-step solution

1. 1

   Use the defining relation for radian measure. The arc length formula

   $s=r\theta$

   is in fact the definition of the radian: an angle of $1$ radian cuts an arc equal in length to the radius. Solving for $r$,

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

2. 2

   Substitute the given values.

   $r=\frac{2}{1/3}$

3. 3

   Divide by the fraction. Dividing by $\tfrac13$ means multiplying by $3$:

   $r=2\times 3=6\ \text{ft}$

   $\boxed{r=6.000\ \text{ft}}$

4. 4

   Check by going forwards. With $r=6$ and $\theta=\tfrac13$, the arc is $s=6\times\tfrac13=2$ ft ✓.

5. 5

   Sanity-check the magnitude. Since $\theta<1$ radian, the arc must be shorter than the radius — and $2<6$ ✓. Had the answer come out smaller than the arc length, a reciprocal would have been used the wrong way round.