To calculate the perimeter of the regular hexagon, we first need to find the length of one side, $s$, using the formula for the apothem:

$a = \frac{s}{2 \tan\left(\frac{180^\circ}{n}\right)}$

Where:

• $a$ is the apothem (given as $2\sqrt{3}$),
• $s$ is the length of one side,
• $n$ is the number of sides (which is 6 for a hexagon).

Step 1: Solve for $s$

Rearranging the formula to solve for $s$:

$s = 2a \tan\left(\frac{180^\circ}{n}\right)$

Substituting $a = 2\sqrt{3}$ and $n = 6$:

$s = 2(2\sqrt{3}) \tan\left(\frac{180^\circ}{6}\right)$ $s = 4\sqrt{3} \tan(30^\circ)$

We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}}$, so:

$s = 4\sqrt{3} \times \frac{1}{\sqrt{3}} = 4$

Step 2: Calculate the perimeter

The perimeter of a regular polygon is given by:

$P = n \times s$

For a hexagon with 6 sides, and each side of length 4 units:

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

Thus, the perimeter of the hexagon is 24 units.

Would you like more details or clarifications?

Here are some related questions:

1. How do you find the area of a regular hexagon using the apothem and side length?
2. What is the relationship between the apothem and the radius of a regular hexagon?
3. How does the apothem change if the number of sides increases?
4. Can the formula $a = \frac{s}{2 \tan(180^\circ/n)}$ be applied to all polygons?
5. How would the side length change if the apothem was given in different units?

Tip: The apothem of a regular polygon always bisects its side and forms a right triangle with half the side length.