Question

Find the exact perimeter (in centimeters) and area (in square centimeters) of the sector shown.

Answer 1

perimeter: 36 + 12 cm

area: 108 cm²

Explanation:

The first thing we must do is calculate the circumference (the perimeter is the sum of all sides, that is, the arc plus twice the radius) and area of the complete circle, using the following formulas:

`\begin{gathered} c=2\pi r=(2)(18)\pi=36\pi\text{ cm} \\ \\ A=\pi r^2=(18)^2\pi=324\pi\text{ cm}^2 \end{gathered}`

Now, we know that the angle of the sector is 120° and that of the complete circle is 360°, therefore we make the following proportion to calculate in each case:

`\begin{gathered} arc_(120)=c\cdot(120)/(360)=36\pi\cdot(120)/(360)=12\pi\text{ cm} \\ \text{ } \\ p_S=2r+arc_(120)=2\cdot18+12\pi=36+12\pi\text{ cm} \\ \\ A_S=324\pi\cdot(120)/(360)=108\pi\text{ cm}^2 \end{gathered}`

Therefore, the perimeter of the sector is 36 + 12 centimeters and the area of the sector is 108 square centimeters