M. Berger, Geometrie, vol. V. MR0536874

Edit. Let me sketch a proof for the spherical triangle. Let the sphere have area $4\pi$.
First you derive the area of diangle. It is $2\alpha$, where $\alpha$ is the angle,
by completely elementary reasons. Now consider a triangle. Extend its sides to three
full great circles. These three circles make several diangles and two equal triangles
(the second one is centrally symmetric to the original one).
Make a picture showing how these three circles partition the sphere. As the areas of all
diangles are known the area of a triangle is simply derived by the exclusion-inclusion formula!

Notice: this proof is truly elementary in the sense that it only uses the existence of the area
for a diangle and triangle, its invariance with respect to rotations, and finite additivity. Euclid COULD give a rigorous
proof of this. As rigorous as his investigation of areas of Euclidean triangles.