From what i understand, one argument for a flat universe is the fact that the sum of the angles of a cosmic triangle is more or less equal to 180 degrees. What i don’t understand is how we calculate the angles. Most of what I’ve read online state that since we can measure the 3 sides of the cosmic triangle, we can use trigonometry to calculate the angles. But doesn’t the fact that we are using trigonometry to calculate the angles already presume the universe is flat rather than proofs it?

In: Planetary Science

There are two main tests we can do to observe the curvature of the universe.

1. The angular size of the temperature fluctuations that appear in the CMB (Cosmic Microwave Background), which is the remnant radiation of the Big Bang. The fluctuations have a particular spectrum between colder and hotter on specific distance scales (which changes depending on the curvature of the universe). We can observe which scale it follows.

2. The angular separations between galaxies that cluster at different epochs throughout the universe. This one I’m not as familiar with, but I think there’s also a specific distance scale that varies according to the curvature of the universe.

According to these two tests, there is a highly likely probability that the universe is flat. Of course, it’s possible that what we’ve tested is actually at a tiny scale compared to a gigantic spherical universe, making its surface appear flat. It’s just that according to what we can currently observe right now, the universe is flat.

>But doesn’t the fact that we are using trigonometry to calculate the angles already presume the universe is flat rather than proofs it?

No, the math works to calculate the angles regardless of the curvature of the underlying surface. What changes is the **sum of the angles** in a triangle. On a flat surface, the angles sum to 180 degrees. On a surface that is sphere-like, the angles sum to _more_ than 180 degrees. And on a surface that curves the other direction, the sum is _less_ than 180 degrees.

So we can calculate the angles based on the side lengths, and then when we add the angles, that sum tells us something about the curvature.

Trigonometry works in other geometries than flat. So no the fact that you use trigonometry doesnt mean you presume the universe is flat. [https://en.wikipedia.org/wiki/Generalized_trigonometry](https://en.wikipedia.org/wiki/Generalized_trigonometry)

But even if it would presume the universe is flat and trigonometry would only work in a flat space the fact that it works would also be prove that the universe is flat (to the degree that we can measure it). This would be a prove by contradiction.

You are completely correct that knowing three sides is not at all enough. You either need to know the curvature already to get the angles (pointless!) or get one of the angles in a different way. Luckily we can: we usually take Earth as one of the 3 vertices, which means we can just measure the angle we see between the other two.

And indeed, from the 3 side lengths, 3 angles and the curvature (or alternatively the area) any 4 determine the rest, but 3 are _not_ enough.

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