Question

find the arean of the regular polygon. round your answer to the nearest tenth

Answer 1

The area of the hexagon is given as:

$A=\frac{1}{2}Pa$

where P is the perimeter and a is the apothem.

Now, we need to notice that the we have the following triangle:

and we notice that x is half the length of each side of the hexagon.

From this triangle we notice that:

$\begin{gathered} \tan 60=\frac{6}{x} \\ x=\frac{6}{\tan60} \\ x=\frac{6}{\sqrt[]{3}} \\ x=\frac{6\sqrt[]{3}}{3} \end{gathered}$

Once we found x that means that each side has length :

$\frac{12\sqrt[]{3}}{3}$

Now, the perimeter is:

$6\cdot\frac{12\sqrt[]{3}}{3}=24\sqrt[]{3}$

Plugging this values and the apothem in the area formula we have that:

$A=\frac{1}{2}(24\sqrt[]{3})(6)=72\sqrt[]{3}=124.7$

Therefore the area is 124.7 squared inches