Arcs and Central Angles

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A central angle is an angle whose vertex is the center of a circle. The arc inside a central angle has the same measure as the angle. Arcs and central angles let you measure parts of a circle in degrees, where the whole circle is 360°.

Before this lesson: You should know the Parts of a Circle, Measuring and Classifying Angles and the Segment Addition Postulate.

Central Angles and Arcs

An arc is a part of a circle between two points. A central angle has its vertex at the center, and its sides are radii. The arc between the two radii is the arc the angle intercepts.

Circle with center O. Radii OA and OB form a central angle of 64 degrees. The minor arc AB, drawn thick in orange, also measures 64 degrees. The rest of the circle, major arc ACB through point C, is drawn in blue and measures 360 minus 64, or 296 degrees.
Central angle AOB is 64°, so minor arc AB is 64° and major arc ACB is 296°.

A full circle is 360°. Each central angle cuts the circle into two arcs.

Type of arcMeasureHow to name it
Minor arcLess than 180°, equal to its central angleTwo letters: \overset{\frown}{AB}
Major arcMore than 180°: 360° minus the minor arcThree letters: \overset{\frown}{ACB}
SemicircleExactly 180°, cut off by a diameterThree letters

The middle letter in a three-letter name is any point on the arc. It tells you which way around the circle to go. Write the measure of an arc as m\overset{\frown}{AB}.

The Arc Rules

Central angle rule: The measure of a minor arc equals the measure of its central angle.

m\overset{\frown}{AB} = m\angle AOB

Major arc rule: A major arc and its minor arc add to 360°.

m\overset{\frown}{ACB} = 360^\circ - m\overset{\frown}{AB}

Arc Addition Postulate: The measure of an arc made of two arcs that share only an endpoint is the sum of the two arcs. It works like the segment addition postulate.

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

Congruent arcs and chords: In the same circle, or in congruent circles, two arcs are congruent when their central angles are congruent. Congruent arcs also have congruent chords, and the reverse is true.

How to Find Arc Measures

  1. Find the central angle that intercepts the arc. A minor arc has the same measure.
  2. For a major arc, subtract the minor arc from 360°.
  3. For an arc made of smaller arcs, add them.
  4. If angles around the center are given as expressions, set their sum equal to 360° (whole circle) or 180° (semicircle), then solve.

Worked Examples

Example 1: Minor and major arcs

In circle O, m\angle AOB = 64^\circ. Find m\overset{\frown}{AB} and m\overset{\frown}{ACB}.

  1. A minor arc equals its central angle: m\overset{\frown}{AB} = 64^\circ
  2. Write the major arc rule: m\overset{\frown}{ACB} = 360^\circ - 64^\circ
  3. Subtract: m\overset{\frown}{ACB} = 296^\circ

Answer: m\overset{\frown}{AB} = 64^\circ and m\overset{\frown}{ACB} = 296^\circ

Example 2: Arc Addition Postulate

Points A, B and C are on circle O, with B between A and C. m\overset{\frown}{AB} = 70^\circ and m\overset{\frown}{BC} = 45^\circ. Find m\overset{\frown}{ABC}.

  1. Write the Arc Addition Postulate: m\overset{\frown}{ABC} = m\overset{\frown}{AB} + m\overset{\frown}{BC}
  2. Substitute: m\overset{\frown}{ABC} = 70^\circ + 45^\circ
  3. Add: m\overset{\frown}{ABC} = 115^\circ

Answer: m\overset{\frown}{ABC} = 115^\circ

Example 3: Central angles on a diameter

\overline{AD} is a diameter of circle O. Points B and C are on the same side of the diameter. m\angle AOB = 3x + 10, m\angle BOC = 5x and m\angle COD = 2x + 20. Find each angle.

Circle with center O and diameter AD. Points B and C lie on the upper semicircle. The central angles are angle AOB equals 3x plus 10, angle BOC equals 5x, and angle COD equals 2x plus 20. They add to 180 degrees, so x is 15 and the angles are 55, 75 and 50 degrees.
The three angles fill a straight angle at O, so they add to 180°.
  1. The three angles make a straight angle at O, so they add to 180°: (3x + 10) + 5x + (2x + 20) = 180
  2. Combine like terms (see combining like terms): 10x + 30 = 180
  3. Subtract 30 from both sides: 10x = 150
  4. Divide both sides by 10: x = 15
  5. Find m\angle AOB: 3(15) + 10 = 45 + 10 = 55^\circ
  6. Find m\angle BOC: 5(15) = 75^\circ
  7. Find m\angle COD: 2(15) + 20 = 30 + 20 = 50^\circ
  8. Check: 55 + 75 + 50 = 180

Answer: The angles are 55°, 75° and 50°. So m\overset{\frown}{AB} = 55^\circ, m\overset{\frown}{BC} = 75^\circ and m\overset{\frown}{CD} = 50^\circ.

Example 4: Word problem with a pie chart

In a survey, 40% of students chose pizza as their favorite lunch. What central angle should the pizza slice of a pie chart have?

  1. The whole circle is 360°, and the slice is 40% of it: 0.40 \times 360^\circ
  2. Multiply: 144^\circ

Answer: The pizza slice has a central angle of 144°, so its arc is 144°. Every pie chart works this way (see percentages).

Example 5: Word problem with a clock

At 4:00, what is the measure of the smaller arc between the hour hand and the minute hand? What is the larger arc?

  1. The 12 numbers split the circle into 12 equal arcs: 360^\circ \div 12 = 30^\circ
  2. At 4:00, the hands are 4 numbers apart: 4 \times 30^\circ = 120^\circ
  3. Find the larger arc: 360^\circ - 120^\circ = 240^\circ

Answer: The smaller arc is 120° and the larger arc is 240°.

Common Mistakes

MistakeExample of the errorFix
Naming a major arc with two lettersWriting \overset{\frown}{AB} for the 296° arcUse three letters for a major arc: \overset{\frown}{ACB}.
Subtracting from 180° for a major arc180^\circ - 64^\circ = 116^\circA whole circle is 360°: 360^\circ - 64^\circ = 296^\circ.
Adding arcs that overlapAdding \overset{\frown}{AC} and \overset{\frown}{BD} when they share a pieceOnly add arcs that meet at a single endpoint.
Thinking equal degrees means equal lengthSaying a 60° arc on a big circle is as long as a 60° arc on a small circleArc measure is in degrees. Length depends on the radius (see arc length).
Using 360° for angles on a diameterSetting angles on one side of a diameter equal to 360°Angles on one side of a diameter add to 180°.

Practice Problems

  1. In circle O, m\angle AOB = 38^\circ. Find m\overset{\frown}{AB} and the major arc m\overset{\frown}{ACB}.
  2. A major arc measures 250°. Find its minor arc and the central angle of the minor arc.
  3. Arcs \overset{\frown}{AB} and \overset{\frown}{BC} share only point B. m\overset{\frown}{AB} = 85^\circ and m\overset{\frown}{BC} = 120^\circ. Find m\overset{\frown}{ABC}.
  4. \overline{AC} is a diameter, and B is on the circle. m\overset{\frown}{AB} = 2x + 8 and m\overset{\frown}{BC} = 4x - 2. Find x and each arc.
  5. Four central angles fill a circle: 4x, 5x, 6x and 9x. Find each angle.
  6. In circle O, chord AB is congruent to chord CD, and m\overset{\frown}{AB} = 78^\circ. Find m\overset{\frown}{CD}.
  7. A pie chart shows that 25% of a budget is rent and 15% is food. Find the central angle for each slice.
  8. At 2:00, what is the smaller arc between the clock hands?
  9. A wheel has 36 spokes spaced evenly around the hub. What is the central angle between two spokes that are next to each other?
  10. Circle P is small and circle Q is large. Each has a 40° arc. Are the arcs congruent? Explain.
Show answers
  1. A minor arc equals its central angle: m\overset{\frown}{AB} = 38^\circ
    Write the major arc rule: m\overset{\frown}{ACB} = 360^\circ - 38^\circ
    Subtract: m\overset{\frown}{ACB} = 322^\circ
  2. Subtract from 360°: 360^\circ - 250^\circ = 110^\circ
    The minor arc is 110°, and its central angle is also 110°.
  3. Write the Arc Addition Postulate: m\overset{\frown}{ABC} = 85^\circ + 120^\circ
    Add: m\overset{\frown}{ABC} = 205^\circ
  4. A diameter cuts off a semicircle, so the arcs add to 180°: (2x + 8) + (4x - 2) = 180
    Combine like terms: 6x + 6 = 180
    Subtract 6 from both sides: 6x = 174
    Divide both sides by 6: x = 29
    Find m\overset{\frown}{AB}: 2(29) + 8 = 58 + 8 = 66^\circ
    Find m\overset{\frown}{BC}: 4(29) - 2 = 116 - 2 = 114^\circ
    Check: 66 + 114 = 180
  5. The angles fill the circle: 4x + 5x + 6x + 9x = 360
    Combine like terms: 24x = 360
    Divide both sides by 24: x = 15
    Multiply to find each angle: 4(15) = 60^\circ, 5(15) = 75^\circ, 6(15) = 90^\circ, 9(15) = 135^\circ
    Check: 60 + 75 + 90 + 135 = 360
  6. Congruent chords in the same circle have congruent arcs: m\overset{\frown}{CD} = 78^\circ
  7. Rent: 0.25 \times 360^\circ = 90^\circ
    Food: 0.15 \times 360^\circ = 54^\circ
  8. Each hour mark is 360^\circ \div 12 = 30^\circ
    The hands are 2 marks apart: 2 \times 30^\circ = 60^\circ
  9. Divide the circle into 36 equal angles: 360^\circ \div 36 = 10^\circ
  10. No. Both arcs measure 40°, but congruent arcs must be in the same circle or in congruent circles. The arc on the larger circle is longer.

FAQ

Is the arc measure equal to the central angle?

Yes. A minor arc has the same degree measure as the central angle that intercepts it. A major arc is 360° minus that angle.

What is the difference between a minor arc and a major arc?

A minor arc is less than 180° and is named with two letters. A major arc is more than 180° and is named with three letters.

What is the Arc Addition Postulate?

If two arcs share only an endpoint, the measure of the arc they form together is the sum of their measures. For example, 70° + 45° = 115°.

How many degrees are in a semicircle?

A semicircle is half of a circle, so it measures 180°. A diameter cuts a circle into two semicircles.

Summary

  • A central angle has its vertex at the center of the circle.
  • A minor arc equals its central angle. A major arc is 360° minus the minor arc. A semicircle is 180°.
  • Name minor arcs with two letters and major arcs and semicircles with three letters.
  • Arc Addition Postulate: add arcs that share only an endpoint.
  • In the same or congruent circles, congruent central angles give congruent arcs and congruent chords.

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Next lesson: Inscribed Angles

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