Question

a regular hexagon has a radius of 20 in. what is the approximate area of the hexagon?
600 in.²
1,038 in.²
1,200 in.²
2,076 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 \(r = 20\) in, so the side - length of each equilateral triangle \(a=20\) in.

Step2: Find area of one equilateral triangle

The area formula for an equilateral triangle is \(A_{\triangle}=\frac{\sqrt{3}}{4}a^{2}\). Substituting \(a = 20\) in, we get \(A_{\triangle}=\frac{\sqrt{3}}{4}\times20^{2}=\frac{\sqrt{3}}{4}\times400 = 100\sqrt{3}\text{ in}^2\).

Step3: Find area of hexagon

Since the hexagon is composed of 6 such equilateral triangles, the area of the hexagon \(A = 6\times A_{\triangle}\). Substituting \(A_{\triangle}=100\sqrt{3}\text{ in}^2\), we have \(A = 6\times100\sqrt{3}=600\sqrt{3}\text{ in}^2\approx600\times1.732 = 1039.2\text{ in}^2\approx1038\text{ in}^2\).

Answer:

1,038 in²