Clarifying the Constancy of Focal Length in Spherical Concave Mirrors

This paper revisits one of the most fundamental concepts in geometrical optics — the focal length of a concave spherical mirror — and addresses the classroom assertion that it is “not well defined” or “not constant.” Through purely theoretical analysis based on the law of reflection, it demonstrates that the focal length remains a constant geometric quantity only within the paraxial limit, where incident rays make small angles with the principal axis. Beyond this limit, apparent variability arises naturally from spherical aberration and the breakdown of small-angle approximations, without violating the reflection law itself. The study defines this distinction as the Paraxial Paradox: a conceptual conflict between ideal geometry and physical observation. By resolving it, the paper clarifies the precise boundary between constancy and deviation in mirror optics. The work represents an independent student-led theoretical investigation intended to refine classical understanding and highlight the importance of approximation limits in physical laws. It is written for students, educators, and physicists interested in foundational optics and conceptual reasoning.