Calculate rectangle's side length from fitted ellipse and area data

Question

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!

Answer 1

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 $b$.