Question

A sector of a circle of radius 80 miles has an area of $\mathbf{1600}{\mathbf{\text{\hspace{0.17em}miles}}}^{\mathbf{2}}$.Find the central angle (in radians) of the sector.

Short answer

The central angle for the given radius and the circular area is$\frac{\mathbf{1}}{\mathbf{2}}$radian.

Step-by-step solution

Step 1. Given information.

Area $\text{A}=1600{\text{\hspace{0.17em}miles}}^2$ and the radius$r=80$ miles.

Step 2. Write down the concept.

Use the formula of the circular area $\text{A}=\frac{1}{2}{r}^2\theta$.

Step 3. Determining the angle

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

$\theta =2\times \frac{1600}{{80}^2}$

$\theta =\frac{1}{2}$radian.