Chapter 1.2, Problem 72E

To determine

To Write: The area $A$ of an equilateral triangle as the function of its length of its sides $s$ .

Answer to Problem 72E

  $A=\frac{\sqrt{3}}{4}{s}^2$

Explanation of Solution

Given: Area $A$ of a equilateral triangle

The equilateral triangle has the all the three sides equal in length

Thus

In triangle

Using the Pythagoras theorem

Area of triangle = $\frac{1}{2}bh$

Here

Base = $s$

Thus

  $\begin{array}{l}{h}^2=s^2-{\left(\frac{s}{2}\right)}^2\\ solving\\ h^2=s^2-\left(\frac{s^2}{4}\right)\\ h^2=\frac{3}{4}{s}^2\\ h=s\sqrt{\frac{3}{4}}\\ h=s\frac{\sqrt{3}}{2}\end{array}$

Substitute $h=s\frac{\sqrt{3}}{2}$ in area of the triangle

We get

  $\begin{array}{l}A=\frac{1}{2}sh\\ A=\frac{1}{2}s\cdot s\frac{\sqrt{3}}{2}\\ A=\frac{\sqrt{3}}{4}{s}^2\end{array}$

Hence, $A=\frac{\sqrt{3}}{4}{s}^2$