Question

Use Heron's formula to find the area of an equilateral triangle whose side is 'p'.

Answer 1

Length of one side of the equilateral triangle is 'p' units.
Semi-perimeter of a triangle = $s=\frac{p+p+p}{2}$
Semi-perimeter = $\frac{3p}{2}$
Area of a triangle = $\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$
$$\begin{gathered} =\sqrt{\frac{3p}{2}\left(\frac{3p}{2}-p\right)\left(\frac{3p}{2}-p\right)\left(\frac{3p}{2}-p\right)} \\ =\sqrt{\frac{3p}{2}{\left(\frac{p}{2}\right)}^3} \\ =\sqrt{\frac{3p^4}{16}} \\ =\frac{\sqrt{3}{p}^2}{4} \end{gathered}$$
∴ Area of the equilateral triangle found using Heron’s formula is $\frac{\sqrt{3}{p}^2}{4}$ .