There is a beautiful argument, based on the spherical harmonics of the cosmic microwave background, which calculates the curvature of the universe.
Are there any other methods of computing the curvature of the universe?
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The figure below is redrawn by me from Perlmutter, 1998, arxiv.org/abs/astro-ph/ 9812133 and Kowalski, 2008, arxiv.org/abs/0804.4142.
The error ellipse for the CMB observations happens to be oriented in such a way that it provides a very good constraint on spatial flatness. (BTW, "zero curvature of the universe" is kind of wrong -- it's only zero spatial curvature.) But the SN and BAO observations do also constrain the spatial curvature.
I have a longer and reasonably up to date discussion of this sort of thing at the end of section 8.2 in my general relativity book, which is free online.