Let x be the length of the side of the equilateral triangle.
x will also be the base on which the altitude falls.
One side, the altitute and 1/2 the base forms a right triangle.
so
x² = (x/2)² +(11√3)²
x² = (x²/4) +(121)(3)
x² - (x²/4) = (121)(3)
3x²/4 = (121)(3)
x² = (4/3)(121)(3)
x² = 4(121)
x = 2(11)
x = 22