What is the perimeter of the regular hexagon shown in below. if the apothem, XO,
measures 2√3 units? the number of sides is 6

use the formula

a = (s)/(2tan(180/n))

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.

What is the perimeter of the regular hexagon shown in below. If the apothem, XO, measures 2√3 units? The number of sides is 6. Use the formula a = (s)/(2tan(180/n)).

Learn how to calculate the perimeter of a regular hexagon when the apothem is 2√3 units. This solution uses trigonometric concepts and the regular polygon perimeter formula. Suitable for students in Grades 8-10.

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