Calculate square side length

The side of the square $a$ is the square root of the area.

$a = \sqrt{6561} = 81 \text{ cm}$

Calculate square perimeter

Perimeter of square is $4 \times a$.

$P = 4 \times 81 = 324 \text{ cm}$

Find side of regular hexagon

Since perimeter of hexagon ($6s$) equals perimeter of square ($324$), we solve for $s$.

$6s = 324 \implies s = \frac{324}{6} = 54 \text{ cm}$

Calculate hexagon area

Using the area formula for a regular hexagon with side $s=54$.

$\text{Area} = \frac{3\sqrt{3}}{2} \times 54^2 = \frac{3\sqrt{3}}{2} \times 2916 = 4374\sqrt{3} \text{ cm}^2$