Chapter 11, Problem 25SGR

To determine

To find: The area of the hexagon.

Answer to Problem 25SGR

The area of the hexagon is $55.2f{t}^2$ .

Explanation of Solution

Given Information: The side of regular hexagon is $8ft$ .

Calculation: The side of regular hexagon is $8ft$ .

  

Central angle $=\frac{{360}^{\circ }}{6}={60}^{\circ }$

Apothem $\mathrm{co}$ is height of the $\mathrm{AOB}$ it bisect the angle $\mathrm{AOB}$ and the side $\mathrm{AB}$ .

So angle $\mathrm{AOC}=\frac{{60}^{\circ }}{2}={30}^{\circ }$ .

  $\mathrm{AC}=\frac{8}{2}=4\mathrm{ft}$

  $\begin{array}{l}\frac{OC}{AC}=\tan {30}^{\circ }\\ \frac{OC}{4}=\frac{1}{\sqrt{3}}\\ OC=\frac{4}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{4\sqrt{3}}{3}\\ OC=\frac{4\times 1.73}{3}=\frac{6.92}{3}=2.3\end{array}$

So $a=2.3\mathrm{m}$

Perimeter :

  $\begin{array}{l}=6\times \mathrm{side}\\ =6\times 8\\ =48\end{array}$

Area of hexagon:

  $\begin{array}{l}=\frac{1}{2}aP\\ =\frac{1}{2}(2.3)(48)\\ =(2.3)(24)\\ =55.2{\mathrm{ft}}^2\end{array}$

Thus, the area of hexagon is $55.2f{t}^2$ .