Finding the Side of the Square

The area of a square is calculated using the formula: Area = $side^2$

We are given that the area of the square is $2304$ cm$^2$. Let the side length of the square be $s$. Therefore, we have the equation: $s^2 = 2304$ cm$^2$ To find the side length $s$, we need to calculate the square root of the area: $s = \sqrt{2304}$ cm

Upon calculation, the side length of the square is: $s = 48$ cm

Calculating the Perimeter of the Square

The perimeter of a square is found by multiplying the side length by 4: Perimeter = $4 \times side$

Using the side length $s = 48$ cm that we just found: Perimeter of square = $4 \times 48$ cm Perimeter of square = $192$ cm

Determining the Side of the Hexagon

The problem states that the perimeter of the square is equal to the perimeter of a regular hexagon. Perimeter of hexagon = Perimeter of square Perimeter of hexagon = $192$ cm

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

Now, we can set up the equation to find the side length $h$ of the hexagon: $6h = 192$ cm To find $h$, we divide the perimeter by 6: $h = \frac{192}{6}$ cm $h = 32$ cm

Computing the Area of the Hexagon

The area of a regular hexagon with side length $h$ is given by the formula: Area = $\frac{3\sqrt{3}}{2} \times h^2$

Substitute the side length of the hexagon ($h = 32$ cm) into the formula: Area of hexagon = $\frac{3\sqrt{3}}{2} \times (32)^2$ cm$^2$ Area of hexagon = $\frac{3\sqrt{3}}{2} \times 1024$ cm$^2$

Now, simplify the expression: Area of hexagon = $3\sqrt{3} \times \frac{1024}{2}$ cm$^2$ Area of hexagon = $3\sqrt{3} \times 512$ cm$^2$ Area of hexagon = $1536\sqrt{3}$ cm$^2$

Thus, the area of the regular hexagon is $1536\sqrt{3}$ cm$^2$.