We are asked to determine the area swept by the door. We notice that the area forms a sector of a circle. The area of the sector of a circle is given by:
[tex]A=\frac{1}{2}r^2\theta[/tex]
Where:
[tex]\begin{gathered} A=\text{ area} \\ r=\text{ radius} \\ \theta=\text{ angle in radians} \end{gathered}[/tex]
To convert the given angle into radians we use the following conversion factor:
[tex]\pi\text{ radians = 180 degr}ees[/tex]
Now, we multiply by the conversion factor:
[tex]45\times\frac{\pi}{180}=\frac{\pi}{4}[/tex]
Now, we plug in the values in the formula for the area:
[tex]A=\frac{1}{2}(80cm)^2(\frac{\pi}{4})[/tex]
Solving the operations:
[tex]A=800\pi cm^2[/tex]
Therefore, the area is 800 pi square centimeters.
