Step 1: Recall the special area formula.
An equilateral triangle has all three sides equal. For such a triangle there is a neat shortcut for the area that uses only the side length $a$. The formula is $A=\frac{\sqrt{3}}{4}a^2$. We use this because the problem gives us the area and asks about the side, so a formula linking area and side is exactly what we need.
Step 2: Put in the given area.
We are told the area is $4\sqrt{3}\,\text{cm}^2$. So we set the formula equal to this value. \[ \frac{\sqrt{3}}{4}a^2=4\sqrt{3} \]
Step 3: Cancel the common square root.
Both sides have a $\sqrt{3}$, so we can divide both sides by $\sqrt{3}$ to make the numbers simpler. This leaves $\frac{1}{4}a^2=4$.
Step 4: Solve for the side squared.
Multiply both sides by $4$ to clear the fraction. \[ a^2=16 \]
Step 5: Find the side length.
Take the positive square root, because a length cannot be negative. \[ a=\sqrt{16}=4\,\text{cm} \]
Step 6: Build the perimeter.
Perimeter means the total length around the shape. An equilateral triangle has three equal sides, so we just add the side three times, or multiply by $3$. \[ P=3a=3\times 4=12\,\text{cm} \] So the perimeter is $12$ cm, which matches option 4. \[ \boxed{12\ \text{cm}} \]