Three consonants carry one idea: cutting. A judge who cuts a case in two issues a verdict; a knife that cuts gives you a piece, or a carrot. The same letters cut a body into shapely sectors, cut a word from another word (derivation), and, in grammar and maths, cut things into clean categories. One physical gesture, branching into law, food, geometry, and language. Every word here is 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.
The pa'al holds both senses at once: to cut a material and to cut a decision (a court "cuts" a sentence). The nif'al is its passive, to be cut, and, in grammar, to be derived from another word, since a derived word is one "cut from" its source. The remaining binyanim are not lexicalized for this root.
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.