Find the base.

Let b = the length of the base, and h = the height.

Area = 54 = (1/2)bh (for any triangle)

2h - b = 6 (base is 6 less than twice the height)

Rearranging the second of these to solve for b, b = 2h - 6.

Now substituting this in the first equation:
(1/2)(2h - 6)h = 54
(1/2)(2h^2 - 6h) = 54
h^2 - 3h = 54
h^2 -3h - 54 = 0

By any of various methods, this quadratic can be factored into (h-9)(h+6).

h must be the positive root which is 9. Now, substituting back into the formula b = 2h - 6, we have:

b = 2(9) - 6 = 18 - 6 = 12.

Therefore, the base is 12 and the height is 9.