Central angle

High School

Definition

An angle with a vertex on the midpoint of a circle. That is an angle ABC, the points A and C reside on the circles circumference and B is located at the center of the circle. The curve between points A and C is known as the arc A C.

Worked examples

\(\angle AOB = 60^\circ\) with center O and points A, B on the circle

The central angle is formed by two radii; its measure equals the arc AB it intercepts.

\(\)arc \( AC = 120^\circ \implies \angle ABC = 120^\circ\) (B at center)

A central angle's measure in degrees always equals its intercepted arc's measure.

Common mistakes

Placing the vertex on the circumference → Placing the vertex at the center of the circle A central angle's vertex must be at the center; an angle on the circumference is an inscribed angle.
\(\angle ABC = \frac{1}{2} \times \)arc \( AC\) → \(\angle ABC = \)arc \( AC\) Central angles equal their arcs; the ½ relationship applies only to inscribed angles.
Confusing radius length with angle measure → The angle measure depends only on arc size, not radius Two circles can have the same central angle but different radii; angle measures rotation, not distance.

Where you'll use it next

Central angles are foundational for arc length and sector area formulas, appear in circle theorems alongside inscribed angles, and support trigonometry on the unit circle and polar coordinates in precalculus.

Found in 2 StudyPug lessons

Mastering Central and Inscribed Angles in Circles

Geometry

Dive into the world of circle geometry! Explore the fascinating relationships between central and inscribed angles, uncover key properties, and discover practical applications in mathematics and real life.

10th Grade10th

Central Angles and Proofs: Unlocking Circle Geometry