Central Angles and their Arcs

A central angle sits right at the centre of a circle. Its two sides are radii, and between their ends lies an arc that is tied to the angle in the simplest possible way.

Theorem A central angle equals its arc

A central angle has its vertex at the centre and its sides along two radii. Its measure equals the measure of the arc it cuts off — a 1 : 1 relationship.

If the angle at O is θ, the bold arc AB has the same measure:

\angle AOB = \overset{\frown}{AB} = \theta

Theorem Minor arc and major arc

The two radii split the circle into a smaller arc and a larger one. The minor arc equals the angle θ; the major arc is the rest, $360^\circ - \theta$ .

At exactly $180^\circ$ the two arcs are equal — each a semicircle — and the chord joining the endpoints becomes a diameter.

Reference Special central angles

Central angle	Arc	What it is
90°	90°	quarter circle
180°	180°	semicircle — chord is a diameter
270°	270°	three-quarters
360°	360°	the full circle

Example Two quick cases

A 90° central angle

The intercepted (minor) arc equals the angle:

90^\circ

The major arc is the rest:

360^\circ - 90^\circ = 270^\circ

⟹ minor arc 90°, major arc 270°.

From an arc back to the angle

An arc measures

120^\circ

. Its central angle has the same measure.

⟹ the central angle is 120°.

Summary

A central angle has its vertex at the centre, with two radii as its sides.
Its measure equals its intercepted arc exactly: $\angle = \overset{\frown}{arc}$ .
The minor arc is θ and the major arc is $360^\circ - \theta$ ; at 180° they are equal and the chord is a diameter.
Compare: a central angle equals its arc, while an inscribed angle is only half the arc.