Circle Fundamentals

Definition

A circle is a geometric figure consisting of all points in a plane that are at a given distance (the radius) from a given point (the center).

Why It Matters

As the most symmetrical geometric form, the circle’s properties of equidistance and rotational invariance are foundational to physics, optics, and trigonometry.

Core Concepts

• Radius ($r$): The distance from the center to any point on the circle. All radii of a circle are congruent.
• Diameter ($d$): The distance across the circle through the center; $d = 2r$.
  • How to read: “The diameter d equals two r.”
  • Meaning: The diameter is twice the radius—the longest chord, passing through the center.
• Circumference ($C$): The distance around the circle. $C = 2\pi r = \pi d$
  • How to read: “The circumference C equals two pi r or pi d.”
  • Meaning / when to use: Perimeter of a circle. Use $2\pi r$ when radius is known, $\pi d$ when diameter is given.
• Area ($A$): The measure of the surface enclosed by the circle. $A = \pi r^2 = \frac{1}{4}\pi d^2 = \frac{1}{2}Cr$
  • How to read: “The area A equals pi r squared, or one fourth pi d squared, or one half C r.”
  • Meaning: All three forms are equivalent; pick whichever given quantity ($r$, $d$, or $C$) you have.
• Chord: A line segment whose endpoints both lie on the circle.
  • Theorem 6.1.1: A radius perpendicular to a chord bisects the chord.
  • Theorem 6.3.1: A line through the center perpendicular to a chord bisects the chord and its arc.
• Arc ($\overset{\frown}{AB}$): A portion of the circle’s circumference.
  • Minor Arc: Less than a semicircle ($< 180^\circ$).
  • Semicircle: Exactly half the circle ($180^\circ$).
  • Major Arc: More than a semicircle ($> 180^\circ$).
• Secant: A line that intersects the circle at two points.
• Tangent: A line that touches the circle at exactly one point (point of tangency).
• Circle Principles:
  1. Distance from Center: A point is outside, on, or inside a circle based on whether its distance from the center is greater than, equal to, or less than the radius.
  2. Congruence: Radii or diameters of the same or congruent circles are congruent.
  3. Central Angles and Arcs: In the same or congruent circles, congruent central angles have congruent arcs (and vice versa).
  4. Chords and Arcs: In the same or congruent circles, congruent chords have congruent arcs (and vice versa).
  5. Chord Distance: In the same or congruent circles, congruent chords are equidistant from the center (and vice versa).
• Inscribed and Circumscribed Figures:
  • Inscribed Polygon: A polygon whose sides are chords of a circle.
  • Circumscribed Circle: A circle that passes through each vertex of a polygon.
  • Circumscribed Polygon: A polygon whose sides are tangents to a circle.
  • Inscribed Circle: A circle that touches all sides of a polygon.
  • Concentric Circles: Coplanar circles with the same center.

Connected Concepts

• Angles in a Circle
• Tangents
• Secants
• Power of a Point Theorems
• Circle Equations, Circle Graphs
• Trigonometric Functions on the Unit Circle