Resolving the question requires abandoning the search for a single, context-free number. Mathematics is not an arbitrary collection of isolated facts; it is a collection of logical systems constructed on specific axioms.
If you remain inside the clean, rigid boundaries of Euclidean geometry, a circle has zero sides because it lacks vertices and straight segments. If you operate within calculus and view shapes through asymptotic limits, a circle represents an idealized polygon with infinite sides. If you view the shape through topology and common language, it divides space into two sides: inside and outside. Recognizing which framework governs the room is what settles the debate.
