Chapter 12.2, Problem 33PPS

To determine

Find the surface area.

Answer to Problem 33PPS

  $299.14c{m}^2$

Explanation of Solution

Given:

The height of the right hexagonal prism is 9 cm and each base edge is 4 cm.

Formula Used:

The Lateral area L of a prism is

L = Ph , where P is the perimeter of the base and h is the height of the prism .

The Surface area S of a prism is

S = L + 2B , where L is its Lateral area and B is the area of a base.

Calculation:

  

Draw the right angled triangle out and find the apothem:

  

Now, first find the value of x = $\frac{360°}{12}=30°$

Also,

  $\begin{array}{l}\tan x=\frac{2}{a}\mathrm{....................}[\tan \theta =\frac{length\text{ of side opposite to }\theta }{\text{length of side adjacent to }\theta }]\\ \Rightarrow \tan 30°=\frac{2}{a}\\ \Rightarrow a=\frac{2}{\tan 30°}\mathrm{...........}[multiply\text{ each side by }\frac{a}{\tan 30°}]\\ \Rightarrow a=\frac{2}{\left(\frac{1}{\sqrt{3}}\right)}\mathrm{.............}[\tan 30°=\frac{1}{\sqrt{3}}]\\ \Rightarrow a=2\sqrt{3}\end{array}$

Perimeter of the base = P = (length of each side )(number of sides )= (4)(6) = 24 cm.

The area of one section of the hexagon , which is a triangle with base 4 cm and height $2\sqrt{3}$ cm is $\frac{1}{2}(base)(height)=\frac{1}{2}(4)(2\sqrt{3})=6.92c{m}^2$

So, the area of the base = B = $6\times 6.92=41.57c{m}^2$ ……[there are 6 equal triangles ]

So, the surface area of the right hexagon prism of height 9 cm is :

S = Ph + 2B = $24\times 9+2\times 41.57=216+83.138=299.14c{m}^2$