Degree Measure of a Circular Arc
What is a Circular Arc?
An arc is a continuous portion of a circle's circumference — essentially a "curved segment" between two points on the circle.
When two points $A$ and $B$ lie on a circle, they divide the circle into two arcs:
- Minor arc $\overset{\frown}{AB}$ — the shorter arc (less than a semicircle)
- Major arc $\overset{\frown}{ADB}$ — the longer arc (greater than a semicircle)
- Semicircle — exactly half the circle ($180°$)
📐 Formal Definition: Degree Measure of an Arc
Definition: The degree measure of an arc is the measure of the central angle that subtends (intercepts) that arc.
If central angle $\angle AOB = \alpha°$, where $O$ is the center of the circle, then:
$$\text{degree measure of } \overset{\frown}{AB} = \alpha°$$
Key Properties:
| Property | Formula / Value |
|---|---|
| Full circle | $360°$ |
| Semicircle | $180°$ |
| Minor arc | $0° < \overset{\frown}{AB} < 180°$ |
| Major arc | $180° < \overset{\frown}{ADB} < 360°$ |
| Major arc + Minor arc | $\overset{\frown}{AB}_{\text{minor}} + \overset{\frown}{AB}_{\text{major}} = 360°$ |
🔗 Central Angle ↔ Arc Relationship
The central angle and its intercepted arc are equal in degree measure. This is the foundational principle:
This means if you know the central angle, you immediately know the arc measure, and vice versa.
Arc Addition Postulate:
If point $C$ lies on arc $\overset{\frown}{AB}$, then:
$$\overset{\frown}{AC} + \overset{\frown}{CB} = \overset{\frown}{AB}$$