Answer:
[tex]Ratio = \frac{R^2 - r^2 }{ r^2}[/tex]
Step-by-step explanation:
Given
See attachment for circles
Required
Ratio of the outer sector to inner sector
The area of a sector is:
[tex]Area = \frac{\theta}{360}\pi r^2[/tex]
For the inner circle
[tex]r \to radius[/tex]
The sector of the inner circle has the following area
[tex]A_1 = \frac{\theta}{360}\pi r^2[/tex]
For the whole circle
[tex]R \to Radius[/tex]
The sector of the outer sector has the following area
[tex]A_2 = \frac{\theta}{360}\pi (R^2 - r^2)[/tex]
So, the ratio of the outer sector to the inner sector is:
[tex]Ratio = A_2 : A_1[/tex]
[tex]Ratio = \frac{\theta}{360}\pi (R^2 - r^2) : \frac{\theta}{360}\pi r^2[/tex]
Cancel out common factor
[tex]Ratio = R^2 - r^2 : r^2[/tex]
Express as fraction
[tex]Ratio = \frac{R^2 - r^2 }{ r^2}[/tex]
