A Shifting Shape

The best way to explain my solution is with a drawing. Below, I mapped out all of the possible arrangements of sides and right angles. It should go without saying, but these diagrams are NOT to scale! They are purely for "bookkeeping" purposes.

18 cases

Then, in each case, I set out to find the area of the quadrilateral. In some cases, no such quadrilateral could be constructed, and in one case, multiple quadrilaterals could. Below are the results of my calculations (the details of which are left to the reader).

areas found

A remark about the color-coding in the image:

• If I needed to find the length of the fourth side, it's written in purple.
• I wrote the area of each quadrilateral inside itself, in green.
• The cases scribbled out in red can not form quadrilaterals.
• The cases scribbled out in gray are duplicate areas.

Also, a note about the quadrilateral with the "30" inside it: In this case, two quadrilaterals were possible. These quadrilaterals have areas of $\frac{3}{2}(10+\sqrt{7})$ and $\frac{3}{2}(10-\sqrt{7})$ , the sum of which is 30. (There's a nice picture you could draw to show why this is the case, obviating the need to calculate each individual area.)

So therefore

$A=16+\frac{27}{2}+\frac{35}{2}+8\sqrt{3}+9\sqrt{2}+7\sqrt{6}+18+30+(\frac{15}{2}+6\sqrt{2})+(10+6\sqrt{2}),$

which simplifies to

$\frac{225}{2}+21\sqrt{2}+8\sqrt{3}+7\sqrt{6}\approx 173.2,$

so $\lfloor A \rfloor = \boxed{173}$ .