Bridging the Ontological Gap: Redefining the Circle from a "Set of Points" to a "Line" to Resolve Terminological Inconsistency in School Geometry

This article identifies an ontological inconsistency in the standard definition of a circle used worldwide: "a circle is the set of all points equidistant from a given point (the center)." While a set of points is discrete with no inherent length, students are asked to calculate its circumference. This gap leads to cognitive dissonance. We present a comparative analysis of circle definitions in textbooks from 12 countries, supplemented by an analysis of textbooks from Egypt, Greece, and Lebanon. This reveals two coexisting traditions: the dominant "set-of-points" definition and an alternative "line-based" definition. Building on ontological analysis, we demonstrate that a set of points and a line are fundamentally different. To resolve the inconsistency, we propose a revised definition: "A circle is a line (curve) all of whose points are equidistant from a given point (the center)." This definition aligns formal language with visual practice, unifies terminology (perimeter for polygons, circumference for the circle as parallel concepts), and is mathematically rigorous. The article discusses pedagogical implications and offers recommendations for introducing this definition in middle school.