Question

Circle 1, circle 2, and circle 3 have the same center and have radii, respectively, of r1 cm, r2 cm, r3 cm, where r1 < r2 < r3. Let a1 be the area of circle 1, let a2 be the area of the region within circle 2 and outside circle 1, and let a3 be the area of the region within circle 3 and outside circle 2, where all areas are in cm2. What are the values of \small \frac{a_{1}}{a_{2}} and \small \frac{a_{2}}{a_{3}} ?

Answer 1

The first area is simply the are of the innermost circle, so we have

$A_1 = \pi r_1^2$

Then, the region inside circle 2 and outside circle 1 is the difference between the areas of these circles:

$A_2 = \pi r_2^2 - \pi r_1^2 = \pi(r_2^2-r_1^2)$

By the same logic, we have

$A_3 = \pi r_3^2 - \pi r_2^2 = \pi(r_3^2-r_2^2)$

So, the ratios are

$\dfrac{A_1}{A_2} = \dfrac{\pi r_1^2}{\pi(r_2^2-r_1^2)} = \dfrac{r_1^2}{r_2^2-r_1^2}$

And similarly

$\dfrac{A_2}{A_3} = \dfrac{\pi(r_2^2-r_1^2)}{\pi(r_3^2-r_2^2)} = \dfrac{r_2^2-r_1^2}{r_3^2-r_2^2}$