Hi there --
First, recall that the sides of equilateral traingles are all equal, but when we speak of the triangle height, we mean the altitude of the triangle (not any one of its side lengths).
So the formula that we'll be using is A = (1/2)b*h, where
b = base of the triangle; and
h = height of the triangle, which we know to be 6 cm.
Thus, since we know the height, we only need to calculate the base in order to find the area.
The find the base length, recall that when you draw an altitude in an equalateral triangle, you bisect one 60 degree angle into two 30 degree angles. That means that you essentially now have two special 30-60-90 triangles. We know that if the short side is x, then the long side (hypotenuse) is 2x, and the middle side (in this case the height of the triangle) sqrt(3)*x. If we can calculate x, then we can find the base, as it is just half of the base. (See video tutorial below for a better illustration of this.)
Thus, setting sqrt(3)*x = 6, solving for x, we find x = 2*sqrt(3). Thus b = 2x = 4*sqrt(3), and Area = (1/2)*b*h = (1/2)*(4*sqrt(3))*6 = 12*sqrt(3) cm².
To help reinforce your understanding of solving application problems with special 30-60-90 triangles, I found a webpage reference that I've listed it below.
Also, as I mentioned above, I put together a short video tutorial walk-through of a solution to your question, and listed it below as well. It's the first one we've done, so while it's a little rough, we'd certainly appreciate any feedback you have time to give us.