To solve this problem, we need to calculate the perimeter of the regular hexagon, given that the apothem $XO$ is $2\sqrt{3}$ units.

Step 1: Understanding the Apothem and Side Length Relationship in a Regular Hexagon

In a regular hexagon, the apothem (the perpendicular distance from the center to a side) is related to the side length. The relationship can be derived from the fact that the hexagon can be divided into 6 equilateral triangles. The apothem of a regular hexagon is given by:

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

where $s$ is the side length of the hexagon.

Step 2: Solving for the Side Length

We are given the apothem as $2\sqrt{3}$ units, so we can set up the equation:

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

To solve for $s$, multiply both sides by 2:

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

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

$s = 4 \text{ units}$

Step 3: Calculating the Perimeter

The perimeter $P$ of a regular hexagon is simply 6 times the side length, because a hexagon has 6 equal sides. Thus:

$P = 6 \times 4 = 24 \text{ units}$

Final Answer:

The perimeter of the regular hexagon is $24$ units.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the formula for the area of a regular hexagon using the apothem and side length?
2. How can we derive the relationship between the apothem and side length for other regular polygons?
3. What is the relationship between the circumradius and side length of a regular hexagon?
4. How does the perimeter change if the apothem were increased by a factor of 2?
5. Can you calculate the area of the hexagon given the apothem value?

Tip: For regular polygons, the apothem plays a crucial role in calculating both the area and the side length, which can help in determining other properties like the perimeter.