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Can We Derive the Equations of the Circle Without Assuming π?

2 points by Mhd-Gashi ↗ HN
How can the circumference of a circle be derived from first principles, using only right-angled triangle constructions and the sine function — without assuming the value of π at any point?

https://doi.org/10.5281/zenodo.15665567

3 comments

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al2o3cr
How are you finding sin(x) for x in degrees?

sin(x) has a Taylor series with simple coefficients, but only when x is in _radians_.

The coefficients of that series for x in degrees are going to "sneak" an assumed value of pi right back into the process...

discoutdynamite
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.