I seriously doubt its possible at all. This isnt a question of an alternate formula, its about computability and number theory. circumference is = π x D. π is an irrational, transcendental, constant. it cant be constructed with any mathematical tool, other than referring to said equation. any other construct that approaches or approximates it, usually has pi or a circle built in so its self-referential.

All forms like radians which appear to work without pi, do so because they just tokenize it somewhere, and do not evaluate it.