Question

the base of a solid right pyramid is a regular hexagon with a radius of 2x units and an apothem of x√3 units. which expression represents the area of the base of the pyramid? ○ x²√3 units² ○ 3x²√3 units² ○ 4x²√3 units² ○ 6x²√3 units²

Explanation:

Step1: Recall the formula for the area of a regular polygon

The area \( A \) of a regular polygon is given by \( A=\frac{1}{2} \times \text{perimeter} \times \text{apothem} \).

Step2: Find the side length of the regular hexagon

In a regular hexagon, the radius (distance from center to a vertex) is equal to the side length. Given the radius is \( 2x \) units, so the side length \( s = 2x \) units.

Step3: Calculate the perimeter of the regular hexagon

A regular hexagon has 6 sides. So the perimeter \( P=6\times s \). Substituting \( s = 2x \), we get \( P = 6\times2x=12x \) units.

Step4: Substitute values into the area formula

We know the apothem \( a=x\sqrt{3} \) units and perimeter \( P = 12x \) units. Using the formula \( A=\frac{1}{2}\times P\times a \), we substitute the values:
\[
A=\frac{1}{2}\times12x\times x\sqrt{3}
\]

Step5: Simplify the expression

First, simplify \( \frac{1}{2}\times12x = 6x \). Then multiply by \( x\sqrt{3} \):
\[
A=6x\times x\sqrt{3}=6x^{2}\sqrt{3}
\]

Answer:

\( 6x^{2}\sqrt{3} \) units\(^{2}\) (corresponding to the option: \( 6x^{2}\sqrt{3} \) units\(^{2}\))