To solve for the area of the base of the pyramid, we need to find the area of the regular hexagon. Here are the steps:
1. Understand the shapes involved: The base of the pyramid is a regular hexagon. A regular hexagon can be divided into 6 equilateral triangles.
2. Determine the side length of the hexagon: In a regular hexagon, the radius (distance from the center to any vertex) equals the side length of the hexagon.
- Given: The radius of the hexagon is [tex]\(2x\)[/tex] units.
- Therefore, the side length of the hexagon is also [tex]\(2x\)[/tex] units.
3. Determine the apothem: The apothem is the perpendicular distance from the center to any side of the hexagon.
- Given: The apothem is [tex]\(x \sqrt{3}\)[/tex] units.
4. Formula for the area of the hexagon:
- The area of a regular hexagon can be found using the formula:
[tex]\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\][/tex]
5. Calculate the perimeter of the hexagon:
- The perimeter (P) is the sum of all side lengths.
- Since the hexagon has 6 sides and each side is [tex]\(2x\)[/tex] units, the perimeter will be:
[tex]\[
P = 6 \times 2x = 12x \, \text{units}
\][/tex]
6. Calculate the area:
- Substitute the perimeter and the apothem into the area formula:
[tex]\[
\text{Area} = \frac{1}{2} \times 12x \times x\sqrt{3}
\][/tex]
- Simplify the expression:
[tex]\[
\text{Area} = \frac{1}{2} \times 12x \times x\sqrt{3} = 6x^2\sqrt{3}
\][/tex]
Finally, the expression that represents the area of the base of the pyramid is:
[tex]\[
6x^2\sqrt{3} \, \text{units}^2
\][/tex]
So, the correct answer is:
[tex]\[
6 x^2 \sqrt{3} \, \text{units}^2
\][/tex]
