Arc

Dependency: Circle. A derived definition: two points cut the closed curve of a circle into two pieces.

Definition

Let $A$ and $B$ be points on circle $\odot O$.

The points $A$ and $B$ divide the circle into two curves. Each curve is called an arc and is denoted by $\overset{\frown}{AB}$.

An arc is a piece of a circle. With the same endpoints, the straight connection is a chord, while the path along the circle is an arc.

Remarks

• Minor arc, major arc, and semicircle. The shorter piece is the minor arc and the longer is the major arc. If $AB$ passes through the center, the two pieces have equal length and each is a semicircle. The notation $\overset{\frown}{AB}$ usually means the minor arc unless otherwise stated.
• Chords and arcs correspond. A chord subtends an arc. In the same circle, equal chords subtend equal arcs and conversely.
• Central and inscribed angles. The measure of an arc is given by its central angle. An inscribed angle subtending the same arc is half the central angle (Central ∠, arc, chord are pairwise equivalent).
• Arc length is proportional to central angle. A complete circle corresponds to $360^\circ$; an arc takes the same fraction of the circumference as its central angle takes of $360^\circ$.