Calculate side of the square

Given $\text{Area} = 1296$, the side $s = \sqrt{1296} = 36$ cm.

$s = \sqrt{1296} = 36$

Calculate perimeter of square and hexagon

Perimeter of square $P = 4 \times 36 = 144$ cm. Since the perimeters are equal, $6a = 144$, where $a$ is the side of the hexagon.

$a = \frac{144}{6} = 24$

Calculate area of the hexagon

The area of a regular hexagon is given by $\frac{3\sqrt{3}}{2} a^2$. Substituting $a = 24$, we get $\frac{3\sqrt{3}}{2} \times 576 = 3 \times 288 \sqrt{3} = 864\sqrt{3}$. Checking the options again, if the area is 216$\sqrt{3},$then the side would be 12. Let's re-verify the square side: $\sqrt{1296} = 36$. $36 \times 4 = 144$. $144 / 6 = 24$. The calculation $864\sqrt{3}$ is mathematically sound. However, the official answer key provides 216$\sqrt{3}.$

$\text{Area} = \frac{3\sqrt{3}}{2} \times 24^2 = 864\sqrt{3}$