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

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) in, we get (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}). Substitute (A_{\triangle}=4\sqrt{3}\text{ in}^2) into the formula, we have (A = 6\times4\sqrt{3}=24\sqrt{3}\text{ in}^2\approx24\times1.732 = 41.568\approx42\text{ in}^2).

Answer:

42 in.²