Hey julietteeden.

I don't know the terminology used in common differential geometry but the arc-length formalization should give the length and a tensor formulation to translate the vector identities between the curved space and R^3 should give the area (or you could use the vector calculus result by calculating the normal of the surface and calculating a surface integral):

Surface integral - Wikipedia, the free encyclopedia

The angles should have a tensor formulation with regards to the inner product between co-ordinate spaces.