I'm studying the area of hyperbolic surfaces and have reached a proposition that is not understandable.I really appreciate it if you could help me with it.
Proposition: A complete hyperbolic surface F with finite area and geodesic boundary is homeomorphic to a compact surface less a finite set and has area -2πχ + πχ'. (χ is the Euler characteristic of the surface and χ' is the Euler Characteristic of the boundary)
I just don't get why it's not still -2πχ like the compact hyperbolic case with geodesic boundary! Also it's proved that an unbounded complete hyperbolic surface with finite area is homeomorphic to a closed surface less a finite set and has the area -2πχ.
Consider an ideal triangle $T$ in the hyperbolic plane (a triangle where all three vertices are on the circle "at infinity"), where you include the 3 boundary geodesics in $T$. Then $\chi(T)=1$ since $T$ is contractible. Then the formula $-2\pi \chi=-2\pi$ that you happen to like would predict negative area, which is, of course, nonsense. But if you take into account that $\chi(\partial T)=3$, you get the right answer: $Area = 3\pi - 2\pi=\pi$.