Question

Derive a formula for area of equilateral triangle in terms of its side using Heron's formula.

Answer 1

Heron's Formula =

Area = $\sqrt{s(s-a)(s-b)(s-c)}$

Where $s=\frac{a+b+c}{2}$

equilateral triangle have equal side.

$a=b=c\Rightarrow a$

$\therefore s=\frac{a+a+a}{2}=\frac{3a}{2}$

Area $=\sqrt{\frac{3a}{2}(\frac{3a}{2}-a)(\frac{3a}{2}-a)(\frac{3a}{2}-a)}$

$=\sqrt{\frac{3a}{2}\left(\frac{a}{2}\right)\left(\frac{a}{2}\right)\left(\frac{a}{2}\right)}$

$=\sqrt{\frac{3a^4}{16}}=\frac{\sqrt{3}{a}^2}{4}$