Circle Fundamentals

Circle: Concepts & Properties

Master the language of circles — understand its parts, symmetry, and geometric relationships to build a strong foundation.

Key Concepts

A segment from the center to any point on the circle. All radii in the same circle are equal.

A chord through the center; d = 2r. Longest chord; axis of symmetry when extended as a line.

A segment with endpoints on the circle. Diameter is the special longest chord.

A portion of the circle's circumference, measured by central angle or length.

An angle with vertex at the center; measures arc proportion.

A line touching the circle at exactly one point; tangent ⟂ radius at point of tangency.

A line intersecting the circle at two points; contains a chord.

Infinite lines of symmetry: every line through the center is an axis of symmetry.

• All diameters have equal length (2r), and every diameter divides the circle into two semicircles.
• Equal chords subtend equal central angles and equal arcs; conversely, equal arcs correspond to equal chords.
• A perpendicular bisector of any chord passes through the center, enabling center location constructions.
• Congruent circles (equal radii) have equal corresponding arcs and chords under equal central angles.
• Tangent segments from the same external point are equal in length (tangent-tangent theorem).

• Treating diameter as a finite segment but forgetting the entire line through the center is an axis of symmetry.
• Swapping r and d in formulas (e.g., using d in πr² without halving first).
• Confusing arc length (linear) with arc measure (angular); mixing units like cm with degrees.
• Thinking only a few symmetry axes exist; in fact, every line through the center is an axis (infinitely many).

Compute circumference and area from radius/diameter with steps.

Test your understanding of basic circle properties.