An equilateral triangle altitude measurements.

Question

An equilateral triangle has three equal sides and if I take per example 11 as the side of the triangle then $11\times3=33$ for the perimeter.I want to find the height or the altitude of the geometric figure called a triangle. So I came up with the result which is$\sqrt\frac{11\times11\times3}{4}$ or a result $\frac{11\sqrt3}{2}$ Why do we use only $\frac{11\sqrt3}{2}$ as the height?

Answer 1

or, in other words; the altitude is $a \cos \left( \dfrac{\pi}{3}\right)$, where $a$ is the edge length.

You have the $\angle BAC = \frac{\pi}{3}$. By trigonometry, you have: $ \dfrac{h}{a} = \cos{\left(\dfrac{\pi}{3}\right)}$