Question

How do you calculate the height of an equilateral triangle?

Answer 1

Compute the height of an equilateral triangle.

As we know that, the height of an equilateral triangle is divided into two equal right-angled triangles by a straight line drawn from the vertex to the opposite side.

We can see that the triangle is divided into two right-angled triangles in the diagram.

Here, side $a$ are the hypotenuse, height is common for both triangles, and other is equal to $\frac{a}{2}$.

We will use the Pythagorean theorem, and we get,

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

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

$\Rightarrow$ $h^2=a^2-\frac{a^2}{4}$

$\Rightarrow$ $h^2=\frac{3}{4}{a}^2$

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

Hence, the height of an equilateral triangle is $\frac{a\sqrt{3}}{2}$.