TITLE: Triangle area on surfaces of constant curvature
QUESTION [5 upvotes]: I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length.
Of course, the formula can easily be derived from the Gauss–Bonnet formula to be
$$A = \frac{1}{\kappa}(\alpha + \beta + \gamma - \pi)$$
for $\kappa \neq 0$. However, I would like to have an elementary geometric proof.
Does anybody know a reference?

REPLY [7 votes]: All the "elementary derivations of that type" are cheating (it may look nice but it proves nothing). 
The only elementary way to introduce area is adding it as an axiom (which is already kind of cheating).
You have to say that there is a additive area-functional on the set of all polygons.
Then you probably want to prove that this functional is unique (or include it in the same axiom).
It remains to notice that your functional $A$ satisfies the same properties and nothing left to prove.