Question

Find the area of an equilateral triangle of side ‘a’ using heron’s formula.

Answer 1

Hint: First, before proceeding for this, we must know the value of the sides of the triangle a, a and a. Then, we know for using heron’s formula, we must know the value of s in that as $s={\displaystyle \frac{a+b+c}{2}}$.Then, before using heron’s formula, we must know the heron’s formula which area A of the triangle as $A=\sqrt{s(s-a)(s-b)(s-c)}$ to get the final answer.

Complete step-by-step answer:
In this question, we are supposed to find the area of the heron’s formula of an equilateral triangle with side ‘a’.
So, before proceeding for this, we must know the value of the sides of the triangle a, a and a.

Then we know for using heron’s formula, we must know the value of s in that as:
$s={\displaystyle \frac{a+a+a}{2}}$
Now, solve the above expression to get the value of s as:
$s={\displaystyle \frac{3a}{2}}$
Then, before using heron’s formula, we must know the heron’s formula which area A of the triangle as:
$A=\sqrt{s(s-a)(s-b)(s-c)}$
Then, as we have already calculated the value of s as ${\displaystyle \frac{3a}{2}}$.
Now, by using the value of s in the above heron’s formula, we get the value of area of the triangle as:
$A=\sqrt{{\displaystyle \frac{3a}{2}}({\displaystyle \frac{3a}{2}}-a)({\displaystyle \frac{3a}{2}}-a)({\displaystyle \frac{3a}{2}}-a)}$
Now, to get the value of the area of the equilateral triangle, solve the above expression as:
$$\begin{array}{rl}& A=\sqrt{{\displaystyle \frac{3a}{2}}\left({\displaystyle \frac{3a-2a}{2}}\right)\left({\displaystyle \frac{3a-2a}{2}}\right)\left({\displaystyle \frac{3a-2a}{2}}\right)}\\ & \Rightarrow A=\sqrt{{\displaystyle \frac{3a}{2}}\times {\displaystyle \frac{a}{2}}\times {\displaystyle \frac{a}{2}}\times {\displaystyle \frac{a}{2}}}\\ & \Rightarrow A=\sqrt{{\displaystyle \frac{3a^4}{16}}}\\ & \Rightarrow A={\displaystyle \frac{\sqrt{3}}{4}}{a}^2\end{array}$$
So, we get the area of the equilateral triangle as $${\displaystyle \frac{\sqrt{3}}{4}}{a}^2$$.
Hence, the area of the equilateral triangle by heron’s formula is $${\displaystyle \frac{\sqrt{3}}{4}}{a}^2$$.

Note: Now, to solve these types of questions we need to know some of the basic formulas such as discussed above, Heron's formula which is mostly used to calculate the area of the scalene triangle but can be used for any triangle as well. So, the heron’s formula is as:
$s={\displaystyle \frac{a+b+c}{2}}$
$A=\sqrt{s(s-a)(s-b)(s-c)}$.