To find the perimeter of a regular hexagon, we can use the formula:

$\text{Perimeter} = 6 \times \text{side length}$

But to find the side length, we need to use the relationship between the apothem and the side length. For a regular hexagon, the apothem ($a$) can be related to the side length ($s$) using the following formula:

$a = \frac{s\sqrt{3}}{2}$

Given that the apothem $a = 2\sqrt{3}$, we can now solve for the side length:

$2\sqrt{3} = \frac{s\sqrt{3}}{2}$

To solve for $s$, multiply both sides by 2 to eliminate the denominator:

$4\sqrt{3} = s\sqrt{3}$

Now, divide both sides by $\sqrt{3}$:

$s = 4$

Now that we know the side length is 4 units, we can find the perimeter:

$\text{Perimeter} = 6 \times 4 = 24 \, \text{units}$

Thus, the perimeter of the hexagon is 24 units.

Would you like further details or have any questions?

Here are five related questions:

1. How is the side length of a regular polygon related to the apothem?
2. What is the formula for the area of a regular hexagon using the apothem?
3. Can you explain the geometric properties of a regular hexagon?
4. How would the perimeter change if the apothem were doubled?
5. What are other methods to calculate the perimeter of a regular polygon?

Tip: In a regular polygon, the apothem is the perpendicular distance from the center to the midpoint of a side. It's useful for finding both the perimeter and area.