Question

Derive the formula for height and area of an equilateral triangle

Answer 1

Find the Area of the triangle:

$\mathrm{Area}\mathrm{of}\mathrm{Triangle}=\frac{1}{2}\mathrm{x}\mathrm{base}\mathrm{x}\mathrm{height}$

Here, $\mathrm{base}=\mathrm{a}\mathrm{and}\mathrm{height}=\mathrm{h}$

As also perpendicular drawn from a vertex to side bisect the side in equilateral triangle.

Applying Pythagoras theorem in the triangle

$a^2=h^2+{\left(\frac{a}{2}\right)}^2$

$\Rightarrow h^2=a^2–\left(\frac{a^2}{4}\right)$

$\Rightarrow h^2=\left(\frac{3a^2}{4}\right)$

$\Rightarrow$ $h=\frac{\sqrt{3}a}{2}$

Now,

Put the value of ‘$\mathrm{h}$’ in the area of the triangle equation

$\mathrm{Area}\mathrm{of}\mathrm{Triangle}=\frac{1}{2}\mathrm{x}\mathrm{base}\mathrm{x}\mathrm{height}$

$\Rightarrow Area=\frac{1}{2}xax\frac{\sqrt{3}a}{2}$

Or

Thus, Area of Equilateral Triangle = $\frac{\sqrt{3}}{4}{a}^2$