The size of extra dimensions
Solution 1
What is the process, the rule or the method which leads me to understand/calculate the size of these extra dimensions? How can I understand or prove that, taking the String example, the additional dimensions are so small? Where does the number of their magnitude come from?
At the present moment we have no experimental evidence that there exist the extra ( at minimum) 6 space dimensions of string theory . This is what forces theorists to a length for the compactified dimensions, similar to the length of the proposed string:
The size of these compactified dimensions is similar to the length of a string, the Planck scale:
$1.6\times10^{35}m$
This is a very small number, inaccessible to experiment, which sees elementary particles as points, not one dimensional strings. Some phenomonologists have proposed that some of these dimensions are large enough to be accessible to exerimental probes, the so called large extra dimensions. Experiments at LHC are looking for the proposed signatures, with no success at the moment. These proposed large extra dimensions are still smaller than a micrometer.
Second question: is there a way by which it has been proved that our perceived dimensions (the three spatial ones) have magnitude of centimeters, meters or whatever?
Our system of units is axiomatic: we define a mass , a length, and a time unit and keep it consistently within the calculations at hand. So it cannot be proven, it is assumed.
Solution 2
Extra dimensions in general have to be compact, since a fourdimensional description fits the world we perceive and measure (so far) very well. It is this compactness that sets a length scale. Usually, we assume that the four "regular" spacetime dimensions are not compact, i.e. extended infinitely. In higher dimensions concepts like angular momentum, including Spin, work differently.
The size of this length scale is not set a priori, and a mechanism giving a precise prediction for the size of the compact directions would be a major breakthrough. However, a number of effects in the lowenergy description of a theory with extra dimensions depend crucially on this scale  most prominently the spacing of the KaluzaKlein mass tower, but also e.g. the scale of symmetry breaking in GUT models with extra dimensions. How available these effects are to experiments is highly modeldependent, though. For example, if one introduces branelocalized fields, the corresponding particles do not show a KaluzaKlein tower of excitations. Therefore, any limits on "the size of extra dimensions" have to be weighted carefully with the model assumptions.
The size and other properties of the extra dimensions become a dynamical quantity if one incorporates General Relativity, which String Theory does inherently. Since these socalled moduli degrees of freedom are usually unstable, a realistic theory must incorporate some mechanism of modulistabilization. There are a number of mechanisms on the market, each with benefits and detriments of its own.
So, in conclusion
 We do not observe the higher KaluzaKlein states of any particle so far. Even with fully branelocalized matter, the graviton should have KK states and their nonobervation puts an upper limit on the size of extra dimensions that shrinks with higher energies probed. A particularly wellmotivated scale is the GUT scale, where a grand unified theory might be broken in the process of compactification.
 The "regular" spacetime dimensions are considered infinite. It is not fully excluded that we do live in a compact universe of immense radius, though. These theories are not very popular, since a compact universe immediately implies Lorentz violation (most notably a minimum energy for free photons) and the Lorentz symmetry of our universe is extremely welltested!
Note that nothing I wrote depends on the actual UV completion, i.e. my statements are independent of String theory.
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Les Adieux
Updated on November 12, 2020Comments

Les Adieux almost 3 years
Supposing we have a model, like the String one, which predicts (or requires) $N$ spacetimes dimensions, for the precision let's talk only about space dimensions.
What is the process, the rule or the method which leads me to understand/calculate the size of these extra dimensions? How can I understand or prove that, taking the String example, the additional dimensions are so small? Where does the number of their magnitude come from?
Second question: is there a way by which it has been proved that our perceived dimensions (the three spatial ones) have magnitude of centimeters, meters or whatever?

ACuriousMind almost 8 yearsThe shape of the extra dimensions is not exactly predicted by string theory, but if they were large, we'd have seen them by now. I also don't know what you mean by our perceived dimensions having a "magnitude of centimeters, meters or whatever?".

Les Adieux almost 8 years@ACuriousMind Not only for String theory. I was talking in general with an example. Supposing we have a theory which requires N dimensions. How could we determine the magnitude of these dimensions? Will they be small like 10^50 m or will they be large like 10^50 m? Moreover I don't think we could see large dimensions. Think about a virus: it doesn't perceive the bigness o the world, it doesn't "see" our three space dimensions, even if for him are really really large...

Jiminion almost 8 yearsThey basically are just adding dimensions to make their math work. As ACM says, if they are large, we'd see them by now. But since they are small, they are just there to make theories and formulae work.

Qmechanic almost 8 yearsFrom a stringtheoretic perspective, the number and shape of extra dimensions are addressed in e.g. physics.stackexchange.com/q/10126/2451 , physics.stackexchange.com/q/31882/2451 , physics.stackexchange.com/q/10527/2451 , physics.stackexchange.com/q/4972/2451 and links therein.

IamZack almost 8 yearsThe size will always depend on the constraints your model will face. A simple example: If you try to solve the Schrödinger equation for a infinite square well in 1+1 dimensions, you'll see that the radius of the extra dimension is constrained by the energy eingenvalues.

Neuneck almost 8 yearsNote that String theory might as well work in less than 10 Dimensions, if one allows a CFT with nontrivial central charge on the worldsheet. Since we have no clue whatsoever how to extract information about this CFT, we just usually assume the central charge comes purely from geometry. This way, we have at least an idea on how to relate observables to what's going on.

Neuneck almost 8 years"So it cannot be proven, it is assumed."  There are definite consequences of a finite universe, most notably quantization of photon wavelengths. Tests of Lorentz invariance thereby also test the scale of the universe, should it be compact.

anna v almost 8 years@Neuneck well, the OP is asking "why the meter". We just decide on a length and call it meter. Look at the link, it is connected to a specific wavelength

Neuneck almost 8 yearsI understood the OP's question (2) as referring not to the measurement scale, which indeed is simply defined, but instead talking about the actual physical scale, which should be independent of manmade systems of measurement. The question reads just as fine if you replace cm and m with lightyears and GPc!