Math Core

When you cut a pizza from the center, each slice gets a piece of the crust. The angle at the tip of the slice decides how much crust you get. In geometry, that piece of crust is called an arc, and the angle at the center tells you its measure.

Central angles

Definition

Central angle

A central angle of a circle is an angle whose vertex is the center of the circle. Its sides are two radii.

In the figure, $\angle AOB$ is a central angle of $\odot O$ . The two radii $\overline{OA}$ and $\overline{OB}$ split the circle into two pieces, called arcs.

Central angle AOB measures 60°. Minor arc AB measures 60° and major arc ACB measures 300°.

Minor arcs, major arcs and semicircles

Two points on a circle split it into two arcs. We name and measure them this way.

A minor arc is smaller than half the circle. Name it with its two endpoints, like $\overset{\frown}{AB}$ . Its measure equals the measure of its central angle.
A major arc is larger than half the circle. Name it with three letters, the endpoints and a point on the arc between them, like $\overset{\frown}{ACB}$ . Its measure is $360^\circ$ minus the measure of the minor arc.
A semicircle is exactly half a circle. Its endpoints are the ends of a diameter, and it measures $180^\circ$ . Name it with three letters too.

Measuring arcs

$m\overset{\frown}{AB} = m\angle AOB \qquad\qquad m\overset{\frown}{ACB} = 360^\circ - m\overset{\frown}{AB}$

A whole circle measures $360^\circ$ and a semicircle measures $180^\circ$ .

In the figure, $m\angle AOB = 60^\circ$ , so $m\overset{\frown}{AB} = 60^\circ$ and $m\overset{\frown}{ACB} = 360^\circ - 60^\circ = 300^\circ$ .

The measure of an arc is in degrees, not in inches or centimeters. It tells you what fraction of the full turn the arc covers, not how long it is. (You'll find the actual length of an arc in a later lesson.)

Common mistake

Two letters always mean the minor arc. If you want the long way around, you need a third letter. So $\overset{\frown}{AB}$ above is $60^\circ$ , not $300^\circ$ .

Adding arcs

Arcs that share exactly one endpoint are adjacent arcs. Just like adjacent angles, their measures add.

Arc Addition Postulate

If $\overset{\frown}{AB}$ and $\overset{\frown}{BC}$ are adjacent arcs, then

$m\overset{\frown}{ABC} = m\overset{\frown}{AB} + m\overset{\frown}{BC}.$

Worked example: Minor and major arcs

In $\odot O$ , $m\angle POQ = 72^\circ$ . Point $R$ is on the circle, not on the minor arc $PQ$ . Find $m\overset{\frown}{PQ}$ and $m\overset{\frown}{PRQ}$ .

The minor arc has the same measure as the central angle: $m\overset{\frown}{PQ} = 72^\circ$ .

The major arc is the rest of the circle: $m\overset{\frown}{PRQ} = 360^\circ - 72^\circ = 288^\circ$ .

Worked example: Using a diameter

$\overline{AD}$ is a diameter of $\odot O$ . Points $B$ and $C$ lie on the same semicircle, in the order $A, B, C, D$ . If $m\overset{\frown}{AB} = 45^\circ$ and $m\overset{\frown}{BC} = 80^\circ$ , find $m\overset{\frown}{CD}$ .

The arcs $\overset{\frown}{AB}$ , $\overset{\frown}{BC}$ and $\overset{\frown}{CD}$ together make the semicircle, which is $180^\circ$ :

$45^\circ + 80^\circ + m\overset{\frown}{CD} = 180^\circ \quad\Longrightarrow\quad m\overset{\frown}{CD} = 55^\circ.$

Congruent arcs

Two arcs are congruent if they have the same measure and they are in the same circle or in congruent circles. In the same circle, congruent central angles cut off congruent arcs, and the reverse is true too.

Why the extra condition? A $90^\circ$ arc of a small coin and a $90^\circ$ arc of a Ferris wheel have the same measure, but they are clearly not the same size, so they can't be congruent.

Worked example: Central angles with algebra

The central angles of a circle measure $x^\circ$ , $2x^\circ$ , $3x^\circ$ and $4x^\circ$ , and together they go all the way around the center. Find the measure of the largest arc.

Central angles around a point add to $360^\circ$ :

$x + 2x + 3x + 4x = 360 \quad\Longrightarrow\quad 10x = 360 \quad\Longrightarrow\quad x = 36.$

The largest arc matches the largest angle: $4(36) = 144^\circ$ .

Tip

Circle graphs (pie charts) are central angles in action. A category that makes up $p\%$ of the data gets a central angle of $\dfrac{p}{100} \cdot 360^\circ$ . For example, $25\%$ gets $90^\circ$ .