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

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.

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

Discover how to find the perimeter of a regular hexagon when given the apothem length of 2√3 units. This geometry problem involves calculating the side length using the apothem-side length relationship and applying the perimeter formula for hexagons. Suitable for students in Grades 8-10.

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