A sector of a circle of radius 5 miles has area 3 square miles. Find the central angle, rounded to three decimal places.

Step-by-step solution

1. 1

   Solve the area formula for the angle. From $A=\tfrac12 r^{2}\theta$,

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

   The radius enters squared, so using $r$ instead of $r^{2}$ here is a five-fold error in this problem.

2. 2

   Substitute.

   $\theta=\frac{2(3)}{5^{2}}=\frac{6}{25}$

3. 3

   Convert to a decimal.

   $\theta=0.24=0.240\ \text{radians}$

   $\boxed{\theta=\dfrac{6}{25}=0.240\ \text{rad}}$

4. 4

   Note that the answer is dimensionless. An angle in radians is a ratio of two lengths, so the miles cancel: $\text{mi}^2/\text{mi}^2$. That is why no unit is attached to $0.240$ beyond the word "radians".

5. 5

   Check by rebuilding the area, and convert to degrees. $A=\tfrac12(25)(0.24)=12.5\times 0.24=3$ square miles ✓. In degrees, $0.24\times\tfrac{180}{\pi}=13.75^{\circ}$ — a narrow wedge, consistent with a sector covering only $3$ of the full circle's $\pi(25)=78.5$ square miles.