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: Divide hexagon into triangles

A regular hexagon can be divided into 6 equilateral triangles with side - length equal to the radius of the hexagon. Here, the radius \(r = 4\) in, so the side - length of each equilateral triangle \(a=4\) in.

Step2: Find area of one equilateral triangle

The area formula for an equilateral triangle is \(A_{\triangle}=\frac{\sqrt{3}}{4}a^{2}\). Substitute \(a = 4\) into the formula: \(A_{\triangle}=\frac{\sqrt{3}}{4}\times4^{2}=\frac{\sqrt{3}}{4}\times16 = 4\sqrt{3}\text{ in}^2\).

Step3: Find area of hexagon

Since the hexagon is composed of 6 such equilateral triangles, the area of the hexagon \(A = 6\times A_{\triangle}=6\times4\sqrt{3}=24\sqrt{3}\text{ in}^2\). Calculate the approximate value: \(24\sqrt{3}\approx24\times1.732 = 41.568\approx42\text{ in}^2\).

Answer:

42 in.²