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: Find the side length of the regular hexagon

In a right - triangle formed by the radius (\(r = 2x\)) and the apothem (\(a=x\sqrt{3}\)) of the regular hexagon, using the Pythagorean theorem \(l=\sqrt{r^{2}-a^{2}}\). But for a regular hexagon with radius \(r\) and apothem \(a\), the side length \(s\) of the regular hexagon can also be found from the relationship in a 30 - 60 - 90 triangle. Since the radius of a regular hexagon is equal to its side length. So \(s = 2x\) (because in a 30 - 60 - 90 triangle related to the regular hexagon, if the apothem \(a=x\sqrt{3}\) and radius \(r = 2x\), the side length of the regular hexagon \(s\) is equal to the radius).

Step2: Use the formula for the area of a regular polygon

The formula for the area of a regular polygon is \(A=\frac{1}{2}Pa\), where \(P\) is the perimeter and \(a\) is the apothem.
The perimeter of a regular hexagon \(P = 6s\). Since \(s=2x\), then \(P=6\times(2x)=12x\).
The apothem \(a=x\sqrt{3}\).
Substitute \(P = 12x\) and \(a=x\sqrt{3}\) into the formula \(A=\frac{1}{2}Pa\):

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Answer:

\(6x^{2}\sqrt{3}\text{ units}^{2}\) (the fourth option)