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}} ?

(1) A₂ = A₃
(2) A₂+A₃ = 2 A₁

Answer 1

a) A₁ /A₂ = r₁² / (r₂² - r₁²)

b) A₂ /A₃ = (r₂² - r₁²) / (r₃² - r₂²)

Explanation:

We have Circle 1 and area A₁

Area of circle 2 outside circle 1 = A₂

Area of circle 3 outside circle 2 = A₃

On the other hand we have

A₁ = π*r₁² area of circle 1

A₂´ = π*r₂² area of circle 2

A₃´ = π*r₃² area of circle 3

All areas in cm²

a) A₁ /A₂

A₁ = π*r₁²

According to problem statement A₂ = π*r₂² - A₁

A₂ = π*r₂² - π*r₁² ⇒ A₂ = π* (r₂² - r₁²)

Then A₁ /A₂ = π*r₁² / [π* (r₂² - r₁²)]

A₁ /A₂ = r₁² / (r₂² - r₁²)

b) A₂ /A₃

A₂ = π* (r₂² - r₁²)

And

A₃ = π* (r₃² - r₂²)

Therefore

A₂ /A₃ = π* (r₂² - r₁²) / π* (r₃² - r₂²) ⇒ A₂ /A₃ = (r₂² - r₁²) / (r₃² - r₂²)

A₂ /A₃ = (r₂² - r₁²) / (r₃² - r₂²)