I'd like to compute the area of the triangle bounded by the three geodesics that form the triangle $T$ of vertices $(0,1,1),(2,0,2),(0,-1,1)$ on the cone $x^2+y^2=z^2$ in $\Bbb R^3$.
The question seems suited for the Gauss Bonnet theorem, though the Gaussian curvature of the cone is zero, so I don't think you can apply it here.
Then there's the option of pulling back the area on $\Bbb R^2$ and integrating using the standard formula $$A(T) = \int_{x^{-1}(T)} \sqrt{EG-F^2}dudv$$ though to know the shape of $T$ you'd have to compute the euqations of the geodesics that bound it, which would be pretty hard and labourious.
Is there any shortcut to this? Like the cone being locally isometric to $\Bbb R^2$ or something?