The height of an equilateral triangle is 3√3 find the perimeter.

Let the side of the triangle be a cm.
By the Pythagorean theorem, from a right-angled triangle formed by height, we write the equality:
a (squared) = 3√3 (squared) + a / 2 (squared);
a (squared) = 27 – a (squared) / 4;
a (squared) – a (squared) / 4 = 27;
4а (squared) / 4 – a (squared) / 4 = 27;
3a (squared) / 4 = 27;
3a (squared) = 108;
a (squared) = 36;
a = 6;