Circle Angles: Central and Inscribed

An introduction to circle terminology — radius, chord, arc — and the fundamental relationship between central and inscribed angles: the central angle is always twice the inscribed angle subtending the same arc.

Keywords: circle, central angle, inscribed angle, arc, chord, radius

Difficulty: beginner

Every circle relates two kinds of angles over the same arc $PQ$: a central angle $\angle POQ$ at the center and an inscribed angle $\angle PRQ$ with vertex on the circle.

Strategy: introduce the circle, its radii, a chord, and an arc, then build the central and inscribed angles subtending that arc and compare them — the central angle is always twice the inscribed one.

The Circle, Center, and Radius

A circle is defined by its center $O$ and radius. Every point on the circle is the same distance from $O$.

A radius is any segment from $O$ to a point on the circle.

Chord and Arc

A chord is a segment connecting two points on the circle. Here $PQ$ is a chord.

An arc is the portion of the circle between two points. The chord $PQ$ divides the circle into two arcs: a minor arc (shorter) and a major arc (longer).

The Central Angle

A central angle has its vertex at the center $O$. The central angle $\angle POQ$ is formed by radii $OP$ and $OQ$.

The measure of a central angle equals the measure of the arc it subtends — the arc between the two radii on the side the angle opens toward.

The Inscribed Angle

An inscribed angle has its vertex on the circle. The inscribed angle $\angle PRQ$ is formed by chords $RP$ and $RQ$.

The angle "opens onto" — subtends — the arc $PQ$ on the far side of chord $PQ$ from $R$. That is the same arc the central angle $\angle POQ$ subtends, which is what lets us compare them.

Central Angle $=2\times$ Inscribed Angle

The fundamental relationship:

The central angle is always twice the inscribed angle subtending the same arc. This is the inscribed angle theorem.

We will prove this rigorously in the next lesson.

We have introduced the fundamental objects of circle geometry:

• Radius, chord, and arc
• Central angle at the center $O$
• Inscribed angle with vertex on the circle

The central angle is always twice the inscribed angle subtending the same arc:

This relationship is the foundation for many results in circle geometry, including Thales' theorem and the cyclic quadrilateral theorem.

Notes

Circle Angle Basics

A circle is the set of all points at a fixed distance (the radius) from a center point $O$.

Key Vocabulary

• Radius: A segment from the center $O$ to any point on the circle.
• Chord: A segment connecting two points on the circle.
• Arc: A portion of the circle between two points.
• Central angle: An angle at the center $O$ whose sides pass through two points on the circle.
• Inscribed angle: An angle whose vertex lies on the circle and whose sides pass through two other points on the circle.
• Subtend: An angle subtends an arc when the angle "opens onto" it — the two sides of the angle pass through the arc's endpoints, and the arc lies across the opening from the vertex.
  • A central angle $\angle POQ$ subtends the arc $PQ$ its radii point toward.
  • An inscribed angle at $R$ subtends the arc $PQ$ on the far side of chord $PQ$ — the arc that does not contain $R$. (Some books say "intercepts" instead of "subtends"; they mean the same thing.)