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 the hexagon into six equilateral triangles

A regular hexagon can be divided into six equilateral triangles with side length \(s = 20\) in (since the radius of a regular hexagon is equal to its side length).

Step2: Find the height of an equilateral triangle

For an equilateral triangle with side length \(s = 20\), the height \(h\) can be found using the Pythagorean theorem. The height \(h=\sqrt{20^{2}-10^{2}}=\sqrt{400 - 100}=\sqrt{300}\approx17.32\) in.

Step3: Calculate the area of one equilateral triangle

The area of a triangle is \(A_{\triangle}=\frac{1}{2}\times base\times height\). Here, base \(b = 20\) in and height \(h\approx17.32\) in. So \(A_{\triangle}=\frac{1}{2}\times20\times17.32 = 173.2\) in.²

Step4: Calculate the area of the hexagon

Since the hexagon is composed of six such triangles, \(A = 6\times A_{\triangle}\). So \(A=6\times173.2=1039.2\approx1038\) in.²

Answer:

1,038 in.²