Question

find the area of the sector. complete each line of the solution by moving one answer to each box. $a=\frac{1}{2}(quad)^2(quad)$ $a = (quad)\text{ in.}^2$

Explanation:

Step1: Recall sector - area formula

The formula for the area of a sector of a circle is $A=\frac{1}{2}r^{2}\theta$, where $r$ is the radius of the circle and $\theta$ is the central - angle in radians. First, convert the angle from degrees to radians. We know that to convert degrees to radians, we use the conversion factor $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. Given $\theta_{deg} = 200^{\circ}$, then $\theta=200\times\frac{\pi}{180}=\frac{10\pi}{9}$ radians, and $r = 4$ inches.

Step2: Substitute values into the formula

Substitute $r = 4$ and $\theta=\frac{10\pi}{9}$ into the formula $A=\frac{1}{2}r^{2}\theta$. So $A=\frac{1}{2}(4)^{2}(\frac{10\pi}{9})$.

Step3: Calculate the area

First, $(4)^{2}=16$. Then $\frac{1}{2}\times16\times\frac{10\pi}{9}= \frac{8\times10\pi}{9}=\frac{80\pi}{9}\approx27.93$ square inches.

Answer:

$A=\frac{1}{2}(4)^{2}(\frac{10\pi}{9})$; $A=\frac{80\pi}{9}\text{ in}^2$