Calculate rectangle's side length from fitted ellipse and area data The problem is the following:

Let there be a rectangle with sides $a$ and $b$ and diagonal $d$. Let
  there be an ellipse with axes $x$ and $y$.
The areas of the two shapes are the same. The ellipse was fitted to
  the rectangle so the centroid of the two shapes is the same, their
  orientation is the same (axes are parallel and "on top of" each
  other). Given $d$, $x$, $y$ and the area, can we calculate $a$ and $b$?

Thanks!

The area (or $x$ and $y$) will give you the product of $a$ and $b$. $d$ will give you the sum of their squares. Using $(a+b)^2= a^2 + b^2 + 2ab$, you can solve for $a$ and $b$. 
