Chapter 11, Problem 27SGR

To determine

To find: The area of the hexagon.

Answer to Problem 27SGR

The area of the hexagon is $100.68{\mathrm{cm}}^2$ .

Explanation of Solution

Given Information: The side of regular hexagon is $6cm$ .

Calculation: The side of regular hexagon is $6cm$ .

  

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

  $\begin{array}{l}\frac{AC}{OA}=\sin {30}^{\circ }\\ \frac{AC}{5}=\frac{1}{2}\\ AC=\frac{5}{2}\\ AC=2.5m\\ AB=2\times AC=2\times 2.5=5\mathrm{cm}\\ \frac{OC}{OA}=\cos {30}^{\circ }\\ \frac{OC}{5}=\frac{\sqrt{3}}{2}\\ OC=5\times \frac{\sqrt{3}}{2}=5\times \frac{1.73}{2}\\ OC=5\times 0.86=4.3m\end{array}$

So $a=4.3m$ .

Perimeter :

  $\begin{array}{l}=6\times \mathrm{side}\\ =6\times 5\\ =30m\end{array}$

Area of hexagon:

  $\begin{array}{l}=\frac{1}{2}aP\\ =\frac{1}{2}(4.3)(30)\\ =(4.3)(15)\\ =64.5m^2\end{array}$

Thus, the area of hexagon is $64.5m^2$ .