How do I find lateral height with only surface area and length of the edge?

I'm guessing you are talking about the surface area of a cone?

Also by length of the edge, do you mean slant height?

Here is the area formula for a cone:

Lateral Area = (π • r • slant height)

Solving for "r" we get:

radius = Lateral Area / (π • slant height)

When you calculate the radius, you can solve for the height by

height^2 = slant height^2 - radius^2