Approximation of Heron's formula

Heron's formula is a formula from antiquity relating the area of a triangle to the lengths of it's 3 sides. It's a nice formula because it works for any 3 sided figure not just right triangles. I was messing around with Heron's formula and I noticed something, if you solve it for one of the sides of the triangle for a particular Area and graph it, it looks like this (I'm only showing where you choose Y to be the second largest number bigger or equal to X)

  This is the graph of the formula for an area of 25 solved for one of the variables. I noticed for most of the solution area the graph is pretty flat. So I approximated it with this:

I did that by putting a plane through the points:

                    

These are valid points on the plane I'm looking for no matter the area A. That took a while to figure out. 

  Finding the plane that goes through those 3 points and simplifying yields:

It is linear in x,y,z for a constant A. That should make a few things more easily possible than with the original Heron's formula without losing much accuracy in the domain of possible triangles.