Solving for Hexagon Area from Square Properties

This problem involves finding the area of a regular hexagon when we know the area of a square and that their perimeters are equal. Let's break down the steps:

Step 1: Finding the Square's Side Length and Perimeter

We are given the area of the square:

Area$_{square}$ = $324 \, \text{cm}^2$

The formula for the area of a square with side length '$s$' is:

Area$_{square}$ = $s^2$

To find the side length '$s$', we take the square root of the area:

$s = \sqrt{\text{Area}_{square}}$

$s = \sqrt{324 \, \text{cm}^2}$

$s = 18 \, \text{cm}$

Now, we find the perimeter of the square. The formula for the perimeter of a square with side length '$s$' is:

Perimeter$_{square}$ = $4s$

Perimeter$_{square}$ = $4 \times 18 \, \text{cm}$

Perimeter$_{square}$ = $72 \, \text{cm}$

Step 2: Finding the Hexagon's Side Length

The problem states that the perimeter of the square is equal to the perimeter of a regular hexagon.

Perimeter$_{hexagon}$ = Perimeter$_{square}$

Perimeter$_{hexagon}$ = $72 \, \text{cm}$

A regular hexagon has 6 equal sides. Let the side length of the hexagon be '$h$'. The formula for the perimeter of a regular hexagon is:

Perimeter$_{hexagon}$ = $6h$

We can now find the side length '$h$' of the hexagon:

$6h = 72 \, \text{cm}$

$h = \frac{72 \, \text{cm}}{6}$

$h = 12 \, \text{cm}$

Step 3: Calculating the Hexagon's Area

The formula for the area of a regular hexagon with side length '$h$' is:

Area$_{hexagon}$ = $\frac{3\sqrt{3}}{2} h^2$

Substitute the value of '$h$' we found:

Area$_{hexagon}$ = $\frac{3\sqrt{3}}{2} (12 \, \text{cm})^2$

Area$_{hexagon}$ = $\frac{3\sqrt{3}}{2} \times 144 \, \text{cm}^2$

Area$_{hexagon}$ = $3\sqrt{3} \times \frac{144}{2} \, \text{cm}^2$

Area$_{hexagon}$ = $3\sqrt{3} \times 72 \, \text{cm}^2$

Area$_{hexagon}$ = $216\sqrt{3} \, \text{cm}^2$

Conclusion

The area of the regular hexagon is $216\sqrt{3} \, \text{cm}^2$. This corresponds to the second option.