Say you have an equilateral triangle of side length . Clearly all the angles are .
If you're expressing the Area in terms of his height, then we need to find an expression for the height. So draw the height, remembering that the height is perpendicular to the base, and will cut one of the angles in half.
So in other words, an equilateral triangle is the same as two right-angled 30, 60, 90 degree triangles put next to each other.
Just looking at one of the right angled triangles, we know that the hypotenuse must be of length and one of the side lengths must be . We can use pythagoras to find the height.
Rearranging for we get .
So now we can say...
Since we get...