Two semicircles inside a rectangle. What is the proportion of the red to the green area?

Scroll down for a solution to this problem.

## Solution

The area red : green is 16 : 81.

We have named the green radius R and the red one r. Then we draw both the line segment connecting the circle centres and the altitude. From the Pythagorean theorem we find that h^{2}=(R+r)^{2}-(R-r)^{2}=4Rr.

This can be rewritten as r/R=(h/2R)^{2}. Now h and 2R are height and width of the rectangle respectively. So r/R=(4/6)^{2}. The requested area proportion is the square of this: (2/3)^{4}=16/81.

## Poem

Geometrical poet Belladonna had another association with this puzzle:

C’est vrai ce sont les couleurs de l’Italie

Mais la comparaison s’arrête ici

La planète verte entre en collision

Avec le soleil rouge en fusion

Les 2 planètes sont choquées

Elles ont un peu trop fêté

2021, la nouvelle année !

Belladonna