A square containing five other squares. What fraction is coloured?

Solution

The coloured fraction is 11/18.

The solution is a matter of careful administrating square sides and congruent triangles. Above the diagonal we marked three congruent right abc-triangles. Below it we marked two congruent right aed-triangles.

Equating the left side of the containing square to the top gives 2a+b=a+2b. This gives a=b. Using the Pythagorean theorem in the abc-triangles, we find that c=a√2.

The bottom side must be 3a, giving e=a/2. Using the Pythagorean theorem in the aed-triangles, we find that d=a√5/2.

Now the total coloured area adds up to 11a²/2, whereas the total area is 9a², leading to a fraction of 11/18.

Poem

The Pandora box
Keeps all pains and tragedies
When you open it
You can find sufferings of the world
And catastrophes
You don’t have to steal the fire Take care !
The Pandora box is made of
Beauty
And passion
And lies
And jealousy
But remember
You can leave it closed !

Bella