Measure the length of one side of the square.

The length of the rectangle will be that length, x, multiplied by a certain constant k you don't know.

So, l = kx

The width will be w = x/k

So that the area of the square is x^2 and the area of the rectangle is (kx)(x/k) = x^2

In your circle of radius r that you measure, you will find that:

Substitute l and w.

Plug in the values of x and r to find k.

From there, you can get l and w, the dimensions of the rectangle to be drawn in your circle.