A central angle of a circle has its vertex at the center. The arc between its sides is the arc it intercepts. Arcs are measured in degrees, by agreement: the measure of an arc is the measure of its central angle. This builds on the parts of a circle from Parts of a Circle.
- Minor arc $\widehat{AB}$: $m\widehat{AB} = m\angle AOB$.
- Major arc $\widehat{ACB}$ (named with a third point on it): $360^\circ - m\widehat{AB}$.
- Semicircle: $180^\circ$.
- Arc Addition Postulate: adjacent arcs add, $m\widehat{ABC} = m\widehat{AB} + m\widehat{BC}$.
Why it works
All circles are similar. Take any two circles, with radii $r$ and $R$. Translate the first so the centers match, then dilate about that center with scale factor $\frac{R}{r}$, as in Properties of Dilations. Every point at distance $r$ lands at distance $R$: the first circle lands exactly on the second. That is why every circle has the same ratio of circumference to diameter, $\pi$, and why an arc's degree measure tells you what fraction of any circle it is.
Key idea
An arc's measure equals its central angle; the whole circle is $360^\circ$. Arc measure is about the angle, not the size of the circle: a $60^\circ$ arc is $\frac{1}{6}$ of any circle.
Worked example: Arcs around a circle
In $\odot O$, $m\angle AOB = 70^\circ$ and $m\angle BOC = 95^\circ$, with $B$ between $A$ and $C$. Find $m\widehat{AB}$, $m\widehat{ABC}$ and the major arc from $A$ to $C$.
- $m\widehat{AB} = 70^\circ$.
- $m\widehat{ABC} = 70 + 95 = 165^\circ$.
- The major arc is $360 - 165 = 195^\circ$.
Answer: $70^\circ$, $165^\circ$, $195^\circ$
Worked example: A semicircle with algebra
$\overline{AC}$ is a diameter, $m\widehat{AB} = (3x + 10)^\circ$ and $m\widehat{BC} = (2x + 20)^\circ$. Find $m\widehat{AB}$.
- The two arcs form a semicircle: $(3x + 10) + (2x + 20) = 180$.
- $5x + 30 = 180$, so $x = 30$ and $m\widehat{AB} = 100^\circ$.
Answer: $100^\circ$
Worked example: Similar circles
What dilation scale factor maps a circle of radius $4$ onto a circle of radius $10$ (after moving the centers together)? How do their circumferences compare?
- $k = \frac{10}{4} = \frac{5}{2}$.
- Every length is multiplied by $\frac{5}{2}$, including the circumference: $8\pi \to 20\pi$.
Answer: $k = \frac{5}{2}$; the larger circumference is $\frac{5}{2}$ times the smaller.
Common mistake
Confusing an arc's measure with its length. Two $60^\circ$ arcs in different circles have the same measure but different lengths; the length depends on the radius. You will find lengths in Arc Length.