Central and inscribed angles

Math

Definition

A central angle has its vertex at the center of a circle and equals its intercepted arc. An inscribed angle has its vertex on the circle and equals half its intercepted arc. This relationship is known as the Inscribed Angle Theorem.

Examples

  • A central angle of 80° intercepts an 80° arc, while an inscribed angle intercepting the same arc measures 40°
  • An inscribed angle that intercepts a semicircle (180° arc) is always 90° — this is Thales' theorem
Key Fact

Inscribed angle = ½ · intercepted arc; Central angle = intercepted arc

Related Terms

Circles and circumference Arc length and sector area Secants and chords Measuring angles

Study This Concept

Practice central and inscribed angles with free review games in these units:

Geometry Unit 9: Circles