The area of a quadrilateral is $(13 - \sqrt{48})\text{ cm}^2$.

1. Set up the equation for the length of the rectangle

$\text{Length} = \frac{\text{Area}}{\text{Width}} = \frac{13 - \sqrt{48}}{2 - \sqrt{3}}$

2. Simplify the surd in the numerator

$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$
$\text{Length} = \frac{13 - 4\sqrt{3}}{2 - \sqrt{3}}$

3. Rationalise the denominator

$\text{Length} = \frac{13 - 4\sqrt{3}}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}}$

4. Expand and simplify to find the final form

$\text{Length} = \frac{26 + 13\sqrt{3} - 8\sqrt{3} - 4(3)}{2^2 - (\sqrt{3})^2}$
$= \frac{26 - 12 + 5\sqrt{3}}{4 - 3}$
$= \frac{14 + 5\sqrt{3}}{1}$
$= 14 + 5\sqrt{3}$

1. Express the area of the square in terms of $c$.

Area $= (c + 2\sqrt{3})^2$
$= c^2 + 2(c)(2\sqrt{3}) + (2\sqrt{3})^2$
$= c^2 + 4c\sqrt{3} + 12$

2. Simplify the given area and equate it to the expanded expression.

Given area $= 13 - \sqrt{48}$
$= 13 - \sqrt{16 \times 3}$
$= 13 - 4\sqrt{3}$

Equating the two expressions for the area:
$c^2 + 12 + 4c\sqrt{3} = 13 - 4\sqrt{3}$

3. Compare rational and irrational parts to find $c$.

Comparing the irrational parts:
$4c\sqrt{3} = -4\sqrt{3}$
$c = -1$

Checking with the rational parts:
$c^2 + 12 = (-1)^2 + 12 = 13$ (consistent)