You're trying to work backwards from notes to intervals, and that's not the way the system works.
Intervals came before notes. People discovered that specific string length (frequency) ratios objectively produced a unique experience that became known as "consonance". Consonance was interesting, because it was rare. Most string pairs didn't produce this experience, only specific ratios. So people figured out what these ratios were, and built instruments with strings that had these ratios. Cool.
The problem is that you can only use one of the strings as the root that the other strings are "intervals of". You also couldn't play your instrument with other people who were tuned to some other arbitrary frequency. Plus the strings don't form intervals between each other, plus you only have "consonance" and no other dissonant notes. Sounds consonant, but not at all functional.
So people decided to create an equally-tempered scale of notes, where the notes were logarithmically linear. Start with the basic multiplier (an octave) of 2. Then split it into n logarithmically-even steps: 2^(1/n). If you set n to 5, you can multiply x by 2^(1/5) to sequentially produce each note of the 5-tet scale, until you get to the fifth which makes it 2^(5/5)x = 2x (the octave).
You can pick any n to cut the octave into n logarithmically even steps (notes), until you get to the octave. So you can split the octave into 4 notes, 5 notes, 6 notes, whatever.
This seems totally disconnected from intervals though. How are logarithmically even scales related to intervals? What's the point of this?
They aren't related. Intervals are for consonance. Equally tempered scales are to produce instruments that are functional. These are two unrelated ideas, but they have to be mashed together in some manner so you can have functional instruments (equal-tempered) but also don't sound like garbage (need the intervals).
So the solution was to just pick arbitrary (literally just guess) values of n, splitting the octave into n steps, and pray that at least a few of the resulting notes have values that match up with your desired intervals (3:2 for example). You also need to pray that the n needed to get your intervals isn't some huge number like 100, because then you'd have 100 notes in your occtave, only 5-6 of which have your desired intervals.
Turns out that no value of n works. There's no n that gives us our desired intervals. The math doesn't work out, and you can prove it. So you never get your desired intervals through equal temperament. But you can pick some values of n where a few of the notes are kinda-sorta-ish-squint-your-ears a bit close approximations of your desired intervals, and then just group them together into a subset-scale you call the "diatonic scale" (this is your maj/min scale), and then shove the rest of the g̶a̶r̶b̶a̶g̶e̶ "accidental" notes off the staff and to the back of the piano.
Turns out 12 is a decent number. We get a few notes that roughly approximate desired intervals, and not too many unnecessary accidental by-product notes. So people stuck with that.
So about the augmented 4th. It's the 6th note. It was never a desired interval. It's an accidental by-product of the 12-TET system. It wasn't produced by finding nice ratios, it was produced by repeatedly multiplying 2^(1/12) to split the octave into 12 logarithmically-even notes (12-Tone Equal Temperament). So it isn't even guaranteed to be a rational number and have a ratio.
How to find it? Just do the 12-TET multiplication:
2^(1/12)^6 = 2^(6/12) = 2^(1/2) = sqrt(2).
Square root of two is irrational. There is no interval ratio.