In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, i.e., $$ S_{\Delta}=\frac{1}{2}a.h, $$ where $a$ is the length of base and the $h$ is the height. Is there a similar formula in Spherical and hyperbolic spaces?
In Euclidean plane, we know that the cosine law says that (suppose $\alpha$, $\beta$, $\gamma$ are the angles, and $a, b, c$ are the lengths opposite $a, b, c$ respectively.) $$ \cos\gamma=\frac{a^2+b^2c^2}{2ab}. $$ Then is there a analogue in spherical and hyperbolic geometry? I have noted that the First and Second Cosine law are not so clearly relevant with this formula.


closed as off topic by Benoît Kloeckner, Bruce Westbury, Anton Petrunin, Ryan Budney, Gerry Myerson Oct 26 '11 at 21:29Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question. 


For hyperbolic geometry, you can find the answers to your questions on Wikipedia. An interesting thing is that there is an absolute upper bound on the area of a hyperbolic triangle, even though lengths are unbounded. For spherical geometry, you can find the area of a spherical triangle here and corresponding trigonometric formulas here. 

