Question

the regular hexagon has a radius of 4 in. what is the approximate area of the hexagon? 24 in.² 42 in.² 48 in.² 84 in.²

Explanation:

Step1: Find the side length and apothem

A regular hexagon can be divided into 6 equilateral triangles. The radius \(r = 4\) in is the side length of the equilateral triangle.
The formula for the apothem \(a\) of a regular hexagon with side length \(s\) (here \(s = r=4\)) is \(a=\frac{\sqrt{3}}{2}s\). So \(a = \frac{\sqrt{3}}{2}\times4=2\sqrt{3}\) in.

Step2: Use the area formula for a regular polygon

The area formula for a regular polygon is \(A=\frac{1}{2}Pa\), where \(P\) is the perimeter and \(a\) is the apothem.
The perimeter \(P\) of a hexagon with side length \(s = 4\) is \(P=6s=6\times4 = 24\) in.
Substitute \(P = 24\) and \(a=2\sqrt{3}\) into the formula: \(A=\frac{1}{2}\times24\times2\sqrt{3}=24\sqrt{3}\approx24\times1.732 = 41.568\approx42\) in².

Answer:

\(42\) in² (the second option)