# Find the central angle θ which forms a sector area 18 square feet of a circle of a radius of 10 feet?

May 20, 2018

The central angle has a measure of $0.36$ radians or $20.626$ degrees.

#### Explanation:

The area of a sector can be determined by the following formula.

$A = \frac{1}{2} {r}^{2} \theta$

$r$ is the radius. $\theta$ is the central angle measure.

In this problem, we want to solve for $\theta$. We can rearrange the formula to make this easier.

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

Now, we just need to plug in the values we know for area and radius.

$\frac{2 \left(18\right)}{10} ^ 2 = \theta$

$\frac{36}{100} = \theta$

$0.36 = \theta$

It is important to remember that this answer is in radians. If we want the answer in degrees, we can convert it.

$\text{radians"="degrees} \cdot \frac{\pi}{180}$

$0.36 = \text{degrees} \cdot \frac{\pi}{180}$

$\frac{64.8}{\pi} = \text{degrees}$

$20.626 = \text{degrees}$