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Central Angles & Arc Measure

A central angle is an angle whose vertex sits at the center of a circle. Its two sides are radii, and they cut off a piece of the circle called an arc. The whole idea of arc measure comes down to one rule: the measure of a minor arc equals the measure of its central angle.

That single rule, plus the fact that a full circle measures 360∘360^\circ, handles every problem in this lesson. If a central angle is 74∘74^\circ, its arc is 74∘74^\circ. If you know one arc, subtract from 360∘360^\circ to get the rest of the circle.

Minor arcs, major arcs, and semicircles

A central angle splits the circle into two arcs. The minor arc is the smaller piece — it measures less than 180∘180^\circ and matches its central angle exactly. The major arc is the larger piece, and its measure is 360∘360^\circ minus the minor arc.

If the central angle is exactly 180∘180^\circ — its sides form a diameter — both arcs are semicircles, each measuring 180∘180^\circ.

Arc measure is in degrees and describes how far around the circle the arc goes. It is not a length in inches or centimeters — two circles of very different sizes can both have a 90∘90^\circ arc.

The arcs of a circle add to 360°

If several central angles fit together around the center with no gaps or overlaps, their arcs cover the whole circle, so the arc measures add to 360∘360^\circ. That's the tool for problems like equal slices of a pizza or a wheel divided into sectors: divide 360∘360^\circ by the number of equal parts.

The same fact works in reverse. If you know all the arcs except one, add the known arcs and subtract from 360∘360^\circ to find the missing one.

Worked examples

Example 1: arc from a central angle

In circle OO, m∠AOB=118∘m\angle AOB = 118^\circ. Find the measure of minor arc ABAB.

A minor arc equals its central anglemAB⌢=m∠AOBm\overset{\frown}{AB} = m\angle AOB
SubstitutemAB⌢=118∘m\overset{\frown}{AB} = 118^\circ

Answer: mAB⌢=118∘m\overset{\frown}{AB} = 118^\circ

Example 2: the major arc

A minor arc measures 65∘65^\circ. Find the measure of its major arc.

The two arcs together make the full circlemajor+minor=360∘\text{major} + \text{minor} = 360^\circ
Subtract the minor arcmajor=360∘−65∘\text{major} = 360^\circ - 65^\circ
Simplifymajor=295∘\text{major} = 295^\circ

Answer: 295∘295^\circ

Example 3: equal parts of a circle

A ferris wheel has 1212 equally spaced cars. What is the arc measure between two neighboring cars?

The full circle measures360∘360^\circ
Divide among the equal spaces360∘12\dfrac{360^\circ}{12}
Simplify30∘30^\circ

Answer: 30∘30^\circ

Try one yourself

74∘74^\circ
OO
AA
BB
??

Common questions

What makes an angle a central angle?

Its vertex is at the center of the circle. If the vertex is on the circle instead, it's an inscribed angle, and a different rule applies — an inscribed angle is only half its arc.

Is arc measure the same as arc length?

No. Arc measure is in degrees and tells you what fraction of the way around the circle the arc goes. Arc length is an actual distance and depends on the radius too. A 90∘90^\circ arc on a huge circle is much longer than a 90∘90^\circ arc on a small one.

How do I know which arc is the minor arc?

The minor arc is the one that measures less than 180∘180^\circ — the shorter way around between the two points. It's usually named with two letters, like arc ABAB. Major arcs get three letters, like arc ACBACB, to show they take the long way around.

Can an arc measure more than 360°?

No. An arc is a piece of one circle, so its measure is between 0∘0^\circ and 360∘360^\circ. If your arcs add to more than 360∘360^\circ, one of them was measured wrong.

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