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

Question

1. In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, then is there a similar thing do happen in Spherical and hyperbolic spaces?

2. 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 1

The amount you turn in radians when going around any shape is precisely 2$\pi$ minus the product of the area of the shape and the amount of curvature. The amount you turn when going around a triangle is the sum of the exterior angles. The sum of the interior angles in radians therefore is $\pi$ plus the product of the area and the curvature.