One idea opens this whole family: opening. Open a door and that is pa'al; the gap you open is an entrance; the tool that opens locks is a key; and when you open up a raw idea over time, you develop it. That last twist is the gift of this root: in Hebrew, development is literally an unfolding, an opening-out. Each word below is a different way of opening, verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
Watch the binyan split the meaning in two: pa'al opens a physical thing, while pi'el לְפַתֵּחַ takes the same opening into the abstract, to develop, to bring out the potential. The nif'al is the passive ("to be opened") and, beautifully, also "to begin": an event opens.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.