Question

Find the area of the sector shown in each figure.

a.

b.

Short answer

a. The area of sector is 2560.

b. The area of sector is 25 square unit.

Step-by-step solution

a. Step 1. Given information. 

$\theta ={80}^{\circ }$i.e. the central angle

Radius of the circle$r=8$

a. Step 2. Write the concept.

Area of$\text{circle}=\pi r^2$

Therefore, Area of sector is:

role="math" localid="1648553849419" $$\begin{gathered} A=\frac{\theta }{2\pi }\pi r^2 \\ =\frac{\theta }{2\pi }\pi r^2 \\ =\frac{1}{2}{r}^2\theta \end{gathered}$$

a. Step 3. Determining the area of sector.

Putting the values in above formula we get:

$$\begin{gathered} A=\frac{1}{2}\times 8^2\times {80}^{\circ } \\ =2560 \end{gathered}$$

b. Step 1. Given information

$$\begin{gathered} \theta =0.5\text{rad} \\ \text{Radius}r=10\text{unit} \end{gathered}$$

b. Step 2. Write the concept.

Area of$\text{circle}=\pi r^2$

Therefore, Area of sector is:

$$\begin{gathered} A=\frac{\theta }{2\pi }\pi r^2 \\ =\frac{1}{2}{r}^2\theta \end{gathered}$$

b. Step 3. Determining the area of sector.

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

We plug the values in the above formula

$$\begin{gathered} A=\frac{1}{2}\times {10}^2\times 0.5 \\ =\frac{100\times 0.5}{2} \\ =\frac{50}{2}\text{squareunit} \\ =25\text{squareunit} \end{gathered}$$