A sector of a circle has central angle 1/3 radian and area 2 square feet. Find the radius, rounded to three decimal places.

Step-by-step solution

1. 1

   Start from the sector-area formula and solve it for r. From

   $A=\frac12 r^{2}\theta$

   multiply by $2$ and divide by $\theta$ before taking the square root:

   $r^{2}=\frac{2A}{\theta},\qquad r=\sqrt{\frac{2A}{\theta}}$

   Rearranging first, rather than substituting into $A=\tfrac12 r^2\theta$ and then wrestling with the numbers, keeps the fraction division in one place.

2. 2

   Substitute the given values.

   $r^{2}=\frac{2(2)}{1/3}=4\times 3=12$

   Dividing by $\tfrac13$ is multiplying by $3$ — the step where a factor of $9$ is easily lost.

3. 3

   Take the positive square root. A radius must be positive, so

   $r=\sqrt{12}=2\sqrt3$

4. 4

   Convert to a decimal.

   $r=2(1.7320508)=3.4641\approx 3.464\ \text{ft}$

   $\boxed{r=2\sqrt3\approx 3.464\ \text{ft}}$

5. 5

   Check by computing the area back. $A=\tfrac12(12)\left(\tfrac13\right)=6\times\tfrac13=2$ square feet ✓. Note the angle must be in radians for this formula; $\tfrac13$ radian is about $19.1^{\circ}$, so the sector is a thin wedge and needs a fairly large radius to reach $2\ \text{ft}^2$.