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Suppose that there is an ellipse that meets with the square, but exactly inside the rhombus. The rhombus's side would be some $x$ cm. (for e.g., we can take it as $2 \ cm$.) The ellipse would have a major axis (with half being $a$) and minor axis (with half being $b$).

What would be the maximum area of the ellipse?

Edit: Rhombus can be freely shaped - it only has limitation of side $x \ cm$ thing.

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    @enzotib Edited t$h$e question.2012-10-02

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A hint: What would be the ellipse of largest area inscribed in a square, and what fraction of the square area would then be covered?

Now given a rhombus of side length $s$ and acute angle $\alpha\leq{\pi\over2}$, what is the maximal area attainable? And for which value $\alpha$ (we are still free to choose it) would this be maximal?