Question

Derive the formula for height and area of an equilateral triangle.

Answer 1

Let $△ABC$ be the triangle where $AP\perp BC,AB=AC=a,BP=PC=\frac{a}{2}$ and $AP=h$ as shown in the above figure:

In $△APC$, $\angle P={90}^0$

Using pythagoras theorem, we have

$A{C}^2=A{P}^2+P{C}^2\Rightarrow a^2=h^2+{\left(\frac{a}{2}\right)}^2\Rightarrow a^2=h^2+\frac{a^2}{4}\Rightarrow h^2=a^2-\frac{a^2}{4}\Rightarrow h^2=\frac{4a^2-a^2}{4}\Rightarrow h^2=\frac{3a^2}{4}\Rightarrow h=\sqrt{\frac{3a^2}{4}}\Rightarrow h=\frac{\sqrt{3}a}{2}$

Now, the area of the triangle is $A=\frac{1}{2}bh$ therefore,

$A=\frac{1}{2}\times b\times h=\frac{1}{2}\times a\times \frac{\sqrt{3}a}{2}=\frac{\sqrt{3}{a}^2}{4}$

Hence, height of an equilateral triangle is $h=\frac{\sqrt{3}a}{2}$ and area is $A=\frac{\sqrt{3}{a}^2}{4}$.