Key takeaway: An arc is a connected piece of a circle’s edge between two points, and its measure in degrees equals the measure of the central angle that cuts it off.
Grade 10 – GeometryG-C.A.2Geometry and Measurement
Definition of Arc
An arc is a curved stretch of a circle, like the crust edge of a pizza slice. Two points on a circle split the edge into two pieces. The shorter piece is the minor one, the longer is the major one, and if the points are opposite ends of a diameter, each piece is a semicircle.
Do not mix up the two ways to size such a piece. Its measure is an angle in degrees, found from the central angle, while its length is a distance along the curve. The two measures belonging to the same two points always total \(360^{\circ}\).
Formula and notation
A minor piece is named with two letters, \(\overset{\frown}{AB}\); a major one uses three, \(\overset{\frown}{ACB}\), where \(C\) is a point on it. For central angle \(\angle AOB\):
\[ m\overset{\frown}{AB} = m\angle AOB \]
The pair satisfies \(m\overset{\frown}{AB} + m\overset{\frown}{ACB} = 360^{\circ}\).
Key facts
- Minor pieces are under 180 degrees, major ones over 180, and a semicircle is exactly 180.
- Measure is in degrees; length is in units like feet.
- Equal measures within one circle give equal chords.
- Neighbors sharing an endpoint add: their measures sum.
Arc example
Points \(A\) and \(B\) on a circle with center \(O\) form a central angle of \(134^{\circ}\). Find the measures of the minor and major pieces, and say what fraction of the circle’s edge the minor one covers.
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The minor piece has the same measure as its central angle: \(m\overset{\frown}{AB} = 134^{\circ}\).
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The major piece is the rest of the circle: \(360 – 134 = 226^{\circ}\).
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Fraction covered by the minor piece: \(\frac{134}{360} = \frac{67}{180}\).
The minor measure is \(134^{\circ}\), the major measure is \(226^{\circ}\), and the minor one covers \(\frac{67}{180}\) of the edge.
Common mistake
Watch out
Noor says that an arc of \(134^{\circ}\) on a circle of radius 12 in. is 134 inches long.
Why it is wrong
Degrees measure the turn, not the distance. The length is a fraction of the circumference: \(\frac{134}{360}\times 2\pi(12) \approx 28.1\) in. Keep measure (degrees) and length (inches) separate.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement