PUMPA - THE SMART LEARNING APP

AI system creates personalised training plan based on your mistakes

Area of an equilateral triangle using Heron's formula:

$$AB$$ $$=$$ $$BC$$ $$=$$ $$CA$$ $$=$$ $$a$$

That is, $$a$$ $$=$$ $$b$$ $$=$$ $$c$$

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

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

Area of a triangle $$=$$ $\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

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

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

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

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

$$=$$ $\frac{a}{2}×\frac{a}{2}\sqrt{3}$

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

Therefore, the area of an equilateral triangle $$=$$ $\frac{\sqrt{3}}{4}{a}^{2}$ sq. units.
Example:
Calculate the area of an equilateral triangle of side $$4 \ cm$$.

$$a$$ $$=$$ $$b$$ $$=$$ $$c$$ $$=$$ $$4 \ cm$$

Area of an equilateral triangle $$=$$ $\frac{\sqrt{3}}{4}{a}^{2}$

$$=$$ $\frac{\sqrt{3}}{4}×\left({4}^{2}\right)$

$$=$$ $\frac{\sqrt{3}}{4}×16$

$$=$$ $$\sqrt{3} \times 4$$

$$=$$ $$1.73 \times 4$$      [Since $$\sqrt{3} = 1.73$$]

$$=$$ $$6.92$$

Therefore, the area of an equilateral triangle $$=$$ $$6.92 \ cm^2$$.