Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry?

2011-10-26T15:14:14Z

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.) $$ \cos\gamma=\frac{a^2+b^2-c^2}{2ab}. $$ 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.

Answer by Tony Huynh for Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry?

2011-10-26T15:37:39Z

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.

Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry? - MathOverflow [closed]

Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry?

Answer by Tony Huynh for Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry?