Question

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

Explanation:

Step1: Recall area formula for regular polygon

The area formula for a regular polygon is $A = \frac{1}{2}aP$, where $a$ is the apothem and $P$ is the perimeter.

Step2: Find side - length of regular hexagon

For a regular hexagon with radius $r = 2x$, the side - length $s$ of the regular hexagon is equal to the radius, so $s=2x$.

Step3: Calculate the perimeter of the hexagon

The perimeter $P$ of a regular hexagon with side - length $s$ is $P = 6s$. Substituting $s = 2x$, we get $P=6\times2x = 12x$.

Step4: Calculate the area of the base

The apothem $a=x\sqrt{3}$. Using the area formula $A=\frac{1}{2}aP$, substitute $a = x\sqrt{3}$ and $P = 12x$. Then $A=\frac{1}{2}\times x\sqrt{3}\times12x=6x^{2}\sqrt{3}$ square units.

Answer:

$6x^{2}\sqrt{3}$ units$^{2}$