Question

the base of a solid right pyramid is a regular hexagon with a radius of 2x units and an apothem of ( xsqrt{3} ) units. which expression represents the area of the base of the pyramid? ( \bigcirc x^2sqrt{3} ) units(^2) ( \bigcirc 3x^2sqrt{3} ) units(^2) ( \bigcirc 4x^2sqrt{3} ) units(^2) ( \bigcirc 6x^2sqrt{3} ) units(^2)

Explanation:

Step1: Recall hexagon area formula

The area of a regular polygon is $\frac{1}{2} \times \text{perimeter} \times \text{apothem}$.

Step2: Find side length of hexagon

For a regular hexagon, side length = radius = $2x$.

Step3: Calculate perimeter

Perimeter $P = 6 \times 2x = 12x$.

Step4: Substitute into area formula

$$\begin{align*} \text{Area} &= \frac{1}{2} \times 12x \times x\sqrt{3} \\ &= 6x \times x\sqrt{3} \\ &= 6x^2\sqrt{3} \end{align*}$$

Answer:

$\boldsymbol{6x^2\sqrt{3}}$ units² (corresponding to the last option)