Central Angles & Arcs

A central angle equals its arc; an inscribed angle is half its arc.

The idea

An arc is a piece of a circle's edge, measured in degrees rather than in length: the full circle is $360°$, and an arc's measure is the share of that full turn it spans, so a semicircle is a $180°$ arc. Two points on a circle cut it into two arcs, the minor (shorter) and the major (longer), whose measures add to $360°$.

Two kinds of angle intercept an arc. A central angle $\theta$ has its vertex at the center $O$ and two radii $r$ for sides; it intercepts the arc $s$ caught between them. An inscribed angle has its vertex at a point $B$ on the circle and two chords for sides; it intercepts the arc across from $B$, running between the chords' far ends $A$ and $C$.

A central angle and its intercepted arc have the same measure — that equality is the definition of degree measure for arcs. An inscribed angle is half its intercepted arc, so an arc is twice any inscribed angle that intercepts it.

Ways to work on it

• Walkthrough. Central angles, inscribed angles, and the arcs they intercept.
• Practice. One quick conversion between angle and arc.
• Hardest. Find unknown angles and arcs when several circle relations combine.

Not sure where to start? Take the ten-question placement test.