How do you find the area of an equilateral triangle?

Question

an equilateral triangle's side is equal to 4z.

I know that the the area is equal to:

 

Answer 1

The height of an equilateral triangle is found by applying Pythagoras' theorem. The perpendicular from a vertex to the opposite side bisects that side, so if the side is a we have a right-angled triangle with side a (the hypotenuse is the side of the equaliteral triangle), and side a/2 the bisected side. The height is the remaining side so (height)^2=a^2-(a/2)^2=(3/4)a^2. The area is half the base times the height=a/2*sqrt(3a^2/4)=a/2*a/2*sqrt(3)=a^2sqrt(3)/4. Your answer uses l instead of a but confirms the answer.