Chapter 12.2, Problem 43HP

To determine

Find the general formula for the total surface area of the prism.

Answer to Problem 43HP

  $\frac{\sqrt{3}}{2}{l}^2+3lh$

Explanation of Solution

Given :

A right prism has a height h units and a base that is an equilateral triangle of side l units.

Calculation:

The base of prism:

  

  $\begin{array}{l}\cos 60°=\frac{\text{length of side adjacent to 60}°}{\text{length of hypotenuse}}=\frac{BD}{AB}\\ \Rightarrow BD=l\cos 60°\\ \Rightarrow BD=\frac{l}{2}\end{array}$

  $\begin{array}{l}\sin 60°=\frac{\text{length of side opposite to 60}°}{\text{length of hypotenuse}}=\frac{AD}{AB}\\ \Rightarrow AD=l\sin 60°\\ \Rightarrow AD=\frac{\sqrt{3}l}{2}\end{array}$

Area of base = B = $\frac{\sqrt{3}}{4}{l}^2$

Now perimeter of base = P = 3l

The Surface area of the prism =

  $\begin{array}{l}2B+Ph\\ =2\frac{\sqrt{3}}{4}{l}^2+3lh\\ =\frac{\sqrt{3}}{2}{l}^2+3lh\end{array}$