Circle

Dependency: Distance. A derived definition: a circle is the set of all points at the same distance from one fixed point.

Definition

Fix a point $O$ in the plane and a positive real number $r$.

The set of all points whose distance from $O$ is exactly $r$ is called the circle with center $O$ and radius $r$, denoted by $\odot O$. The point $O$ is the center, and $r$ is the radius.

In one sentence, a circle is the locus of points equidistant from a fixed point. From this single condition come chords, arcs, tangents, inscribed angles, and the rest of circle geometry.

Remarks

• A circle is a curve, not its interior. Points at distance exactly $r$ form the circle; points at distance less than $r$ are inside it, and points at distance greater than $r$ are outside. The circle together with its interior is a disk.
• Center and radius determine a unique circle. Conversely, every circle has one center and one radius value, so “circle $\odot O$ of radius $r$” is unambiguous.
• Radii, diameters, chords, and arcs are derived from the definition. Every radius has length $r$; a diameter is a longest chord of length $2r$; a chord joins two points on the circle, and an arc is the curve between them.
• Why the radius is positive. For $r=0$ the circle degenerates to a single point, while $r<0$ has no geometric meaning.