What do you mean by conditioning? If you mean numerical conditioning, then it assumes some algorithm to approximate area. Do you have such an algorithm? Then the conditioning is just (which is the error of the estimation).
take the total derivative of S.
this gives you the error as propagated through the area formula for the errors in your 3 parameters.
after a bit of work you come up with
$dS=S\sqrt{(d\gamma)^2\cot^2(\gamma)+\left(\dfrac {da}{a}\right)^2+\left(\dfrac {db}{b}\right)^2}$
the terms with $da$ and $db$ are likely to be small unless the triangle is small with respect to your measuring capabilities. Assuming it isn't we can neglect these 2 terms. That leaves
$\dfrac {dS} S \approx d\gamma \cot(\gamma)$
The problem lies where $\cot(\gamma)$ gets very large, i.e. very small angles, or angles near $\pi$ radians. The limit of both of these is a line segment, not a triangle and so you can see where the problem is.
You should replicate all this so you understand where it comes from.