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 of the hexagon

For a regular hexagon, the radius \(r\) is equal to the side length \(s\). So \(s = 4\) in.

Step2: Find the apothem \(a\)

The formula for the apothem \(a\) of a regular hexagon is \(a=\frac{\sqrt{3}}{2}s\). Substituting \(s = 4\), we get \(a=\frac{\sqrt{3}}{2}\times4 = 2\sqrt{3}\approx3.46\) in.

Step3: Find the perimeter \(P\)

The perimeter \(P\) of a regular hexagon is \(P = 6s\). Substituting \(s=4\), we get \(P=6\times4 = 24\) in.

Step4: Calculate the area \(A\)

The formula for the area \(A\) of a regular polygon is \(A=\frac{1}{2}aP\). Substituting \(a\approx3.46\) and \(P = 24\), we get \(A=\frac{1}{2}\times3.46\times24=41.52\approx42\) in².

Answer:

42 in²