If you draw an altitude down from a vertex to a side, then you get 2 separate right triangles. Both of these right triangles have a hypotenuse of 8, and a leg of 4. To find the other leg (the altitude), you apply the Pythagorean theorem. 4^2 + x^2 = 8^2.
This gets you to x^2 = 48. Because x cannot be negative (as it is a length), x = sqrt(48) = 4 sqrt(3)