Question

How do you calculate the height of an equilateral triangle?

Answer 1

Hint:In the given question, we have been asked how we can calculate the height of any given equilateral triangle. For that, we assume any one side of the triangle to be any variable. Then we cut the equilateral triangle in half and apply the Pythagoras theorem to calculate the height on the terms of the side of the triangle.

Formula Used:
We are going to use the Pythagoras theorem, which is,
\[{\left( {hypotenuse} \right)^2} = {\left( {perpendicular} \right)^2} + {\left( {base} \right)^2}\]

Complete step by step answer:
Let the side of the equilateral triangle be ‘\[a\]’.

Now, we have to calculate the length of \[AD\].
In \[\Delta ADC\], using Pythagoras theorem, we have,
\[{\left( {AD} \right)^2} + {\left( {DC} \right)^2} = {\left( {AC} \right)^2}\]
Now, from the figure,
\[DC = \dfrac{a}{2}\] and \[AC = a\]
Putting the values into the formula, we have,
\[{\left( {AD} \right)^2} + \dfrac{{{a^2}}}{4} = {a^2}\]
Taking all like terms on one side,
\[AD = \sqrt {{a^2} - \dfrac{{{a^2}}}{4}} = \sqrt {\dfrac{{3{a^2}}}{4}} = \dfrac{{a\sqrt 3 }}{2}\]

Hence, \[height = \dfrac{{a\sqrt 3 }}{2}\]

Note: In the given question we had to calculate the height of an equilateral triangle. An equilateral triangle is a triangle with all sides equal. Then we cut the triangle in half. Then we applied the Pythagoras theorem on the half triangle and calculated the height on the basis of the assumed side of the triangle.