How do you calculate the height of an equilateral triangle given the lengths of all 3 sides?

Half the length of one side multiplied by (square root of 3)/(2).

in triangle ABC with height H:

(A•sqrt3)/2 or A (sqrt3)/2

This is because of the Pythagorean theorem.

You draw a line from the top vertex of the triangle vertically to the bottom side. This line is perpendicular to the bottom side and it will bisect that side. Now you want to know the length of your new line. On each side of it, you have a smaller triangle, one side with the length of the side of the original triangle (let's call it s) and one side with length half that, (1/2)s. Since the side with length s will be the hypotenuse of the triangle, we know s2 = (s/2)2 + h2 by the Pythagorean theorem. (h stands for height.)

s2 = s2/4 + h2

s2 (1-1/4) =h2

s2(3/4) =h2

(sqrt3)s/2 = h