farmland, arable land, category
畴 centers on cultivated land: farmland or arable fields. The sense of 'category' likely extends from the idea of dividing land into distinct plots or types.
畴 combines 田 (field) with a phonetic component that historically relates to the sound チュウ. The right side is a simplified form of an older complex character, and the modern shape 寿 is a visual simplification.
The left side 田 is a rice field. The right side 寿 looks like a person measuring and dividing the land. Picture a farmer organizing fields into neat categories of farmland.
For チュウ, imagine a farmer chewing on a straw while surveying the farmland: chew -> チュウ, and the fields stretch out in neat categories.