Given a circle of radius $3\rm{cm}$ inscribed in an equilateral triangle $\triangle ABC$ and $EZDU$ is a square inscribed in the circle.

Question

Given a circle of radius $3\rm{cm}$ inscribed in an equilateral triangle $\triangle ABC$ and $EZDU$ is a square inscribed in the circle. Find the divsion of the area of the square divided by the area of the triangle. (Romania 1961)

The area of the square is obvious from the radios of the circle. We have that $2ZE^2=36\implies ZE^2=18$ hence the area is $18$. This is where I got stuck. I don't know how to calculate the area of the triangle. Could you please explain to me how to solve this question?

Answer 1

$O$ is the centroid of $\triangle ABC$ $\implies OC=2OD$.

Can you take it from here?