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 $\displaystyle \text{alg}(a,b,\gamma)-\dfrac{1}{2}ab\sin(\gamma)$ (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.