Arc Length

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Arc length is the distance along a curved part of a circle. To find it, take the fraction of the circle the arc covers and multiply by the circumference. With the central angle \theta in degrees and radius r, the arc length is \frac{\theta}{360^\circ} \cdot 2\pi r.

Before this lesson: You should know Circumference and Area of Circles, Arcs and Central Angles and Proportions.

Arc Measure vs. Arc Length

An arc has two different numbers. Its arc measure is in degrees, and it equals the central angle. Its arc length is a distance, in units like centimeters or feet.

Two arcs can have the same measure but different lengths. A 60° arc on a pizza is much longer than a 60° arc on a coin. Arc length depends on both the angle and the radius.

The Arc Length Formula

Circle with center O and radius 9. A central angle of 80 degrees cuts off an arc, drawn thick in orange. The arc is 80 out of 360 degrees of the circle, so its length is 80 over 360 times the circumference, 18 pi, which is 4 pi, about 12.57.
The arc is 80/360 of the circle, so its length is 80/360 of the circumference.

The whole circle is 360°, and its length is the circumference, 2\pi r. An arc with central angle \theta is the fraction \frac{\theta}{360^\circ} of the circle. So it is that same fraction of the circumference.

\text{arc length} = \frac{\theta}{360^\circ} \cdot 2\pi r

You can also set up a proportion. The arc is to the whole circle as the angle is to 360°:

\frac{\text{arc length}}{2\pi r} = \frac{\theta}{360^\circ}

Give an exact answer in terms of \pi, like 4\pi. Then use a calculator for a decimal, like 12.57. In Trigonometry you will meet radians, which give a shorter formula, s = r\theta.

How to Find Arc Length

  1. Find the radius. If you know the diameter, divide it by 2.
  2. Find the central angle \theta of the arc. For a major arc, use 360° minus the minor arc.
  3. Write the fraction \frac{\theta}{360^\circ} and simplify it.
  4. Multiply by the circumference, 2\pi r.
  5. Give the exact answer in terms of \pi, then round if asked.

Worked Examples

Example 1: Find an arc length

A circle has a radius of 9 cm. Find the length of an arc with a central angle of 80°. Give the exact answer and the answer rounded to the nearest hundredth.

  1. Write the formula: L = \frac{80}{360} \cdot 2\pi(9)
  2. Simplify the fraction (see fractions): L = \frac{2}{9} \cdot 2\pi(9)
  3. Multiply 2 and 9 in the circumference: L = \frac{2}{9} \cdot 18\pi
  4. Multiply (see multiplying and dividing fractions): L = 4\pi
  5. Use a calculator: L \approx 12.57

Answer: 4\pi cm, or about 12.57 cm

Example 2: Start with the diameter

A circle has a diameter of 20 in. Find the length of a 135° arc.

  1. Find the radius: r = 20 \div 2 = 10
  2. Write the formula: L = \frac{135}{360} \cdot 2\pi(10)
  3. Simplify the fraction: L = \frac{3}{8} \cdot 20\pi
  4. Multiply: L = \frac{60\pi}{8}
  5. Simplify: L = 7.5\pi
  6. Use a calculator: L \approx 23.56

Answer: 7.5\pi in, or about 23.56 in

Example 3: Find the central angle

An arc in a circle of radius 12 m has length 10\pi m. Find its central angle.

  1. Write the formula: 10\pi = \frac{\theta}{360} \cdot 2\pi(12)
  2. Multiply 2 and 12: 10\pi = \frac{\theta}{360} \cdot 24\pi
  3. Divide both sides by 24\pi: \frac{10}{24} = \frac{\theta}{360}
  4. Multiply both sides by 360: \theta = \frac{10}{24} \cdot 360
  5. Multiply: \theta = 150

Answer: The central angle is 150°.

Example 4: Find the radius

A 60° arc has length 6\pi ft. Find the radius of the circle.

  1. Write the formula: 6\pi = \frac{60}{360} \cdot 2\pi r
  2. Simplify the fraction: 6\pi = \frac{1}{6} \cdot 2\pi r
  3. Multiply: 6\pi = \frac{\pi r}{3}
  4. Multiply both sides by 3: 18\pi = \pi r
  5. Divide both sides by \pi: r = 18

Answer: The radius is 18 ft.

Example 5: Word problem with a pendulum

A pendulum is 90 cm long. Its tip swings through an angle of 24°. How far does the tip travel in one swing? Round to the nearest hundredth.

A pendulum 90 centimeters long hangs from a pivot and swings through an angle of 24 degrees. The tip of the pendulum travels along an arc of a circle with radius 90 centimeters. The arc is about 37.70 centimeters long.
The tip moves along an arc with radius 90 cm and central angle 24°.
  1. The tip moves along an arc. The radius is the pendulum length, 90 cm. Write the formula: L = \frac{24}{360} \cdot 2\pi(90)
  2. Simplify the fraction: L = \frac{1}{15} \cdot 2\pi(90)
  3. Multiply 2 and 90: L = \frac{1}{15} \cdot 180\pi
  4. Multiply: L = 12\pi
  5. Use a calculator: L \approx 37.70

Answer: The tip travels about 37.70 cm.

Common Mistakes

MistakeExample of the errorFix
Using the diameter as the radius\frac{135}{360} \cdot 2\pi(20) when the diameter is 20Halve the diameter first: r = 10.
Giving the arc measure as the lengthArc length = 80°Length is a distance: \frac{80}{360} \cdot 2\pi(9) = 4\pi cm.
Using the area formula\frac{80}{360} \cdot \pi(9)^2Arc length uses the circumference, 2\pi r, not the area.
Using the minor arc angle for a major arcUsing 100° for the major arcMajor arc angle = 360^\circ - 100^\circ = 260^\circ.
Rounding \pi too earlyUsing 3 for \piKeep \pi until the end, then use the calculator’s \pi key.

Related Idea: Rolling Wheels

When a wheel rolls one full turn without slipping, it moves forward one circumference, 2\pi r. A part of a turn moves it forward the matching arc length. That is how a bike’s odometer turns wheel turns into distance.

Practice Problems

Give each answer exactly in terms of \pi, then round to the nearest hundredth.

  1. Find the length of a 120° arc in a circle with radius 6 cm.
  2. Find the length of a 45° arc in a circle with radius 10 m.
  3. Find the length of a 270° arc in a circle with diameter 16 in.
  4. An arc of length 5\pi is in a circle of radius 15. Find its central angle.
  5. A 144° arc has length 8\pi. Find the radius.
  6. In a circle of radius 9, a minor arc has a central angle of 100°. Find the length of the major arc.
  7. The minute hand of a clock is 12 cm long. How far does its tip move in 25 minutes?
  8. A bike wheel has a radius of 35 cm. How far does the bike move when the wheel makes 3 full turns?
  9. Find the length of a semicircle with radius 7 ft.
  10. Two circles have radii 5 and 10. Each has a 60° arc. Find both arc lengths. How do they compare?
Show answers
  1. Write the formula: L = \frac{120}{360} \cdot 2\pi(6)
    Simplify the fraction: L = \frac{1}{3} \cdot 12\pi
    Multiply: L = 4\pi
    Use a calculator: L \approx 12.57 cm
  2. Write the formula: L = \frac{45}{360} \cdot 2\pi(10)
    Simplify the fraction: L = \frac{1}{8} \cdot 20\pi
    Multiply: L = 2.5\pi
    Use a calculator: L \approx 7.85 m
  3. Find the radius: r = 16 \div 2 = 8
    Write the formula: L = \frac{270}{360} \cdot 2\pi(8)
    Simplify the fraction: L = \frac{3}{4} \cdot 16\pi
    Multiply: L = 12\pi
    Use a calculator: L \approx 37.70 in
  4. Write the formula: 5\pi = \frac{\theta}{360} \cdot 2\pi(15)
    Multiply 2 and 15: 5\pi = \frac{\theta}{360} \cdot 30\pi
    Divide both sides by 30\pi: \frac{5}{30} = \frac{\theta}{360}
    Multiply both sides by 360: \theta = \frac{5}{30} \cdot 360
    Multiply: \theta = 60^\circ
  5. Write the formula: 8\pi = \frac{144}{360} \cdot 2\pi r
    Simplify the fraction: 8\pi = \frac{2}{5} \cdot 2\pi r
    Multiply: 8\pi = \frac{4\pi r}{5}
    Multiply both sides by 5: 40\pi = 4\pi r
    Divide both sides by 4\pi: r = 10
  6. Find the major arc angle: 360^\circ - 100^\circ = 260^\circ
    Write the formula: L = \frac{260}{360} \cdot 2\pi(9)
    Simplify the fraction: L = \frac{13}{18} \cdot 18\pi
    Multiply: L = 13\pi
    Use a calculator: L \approx 40.84
  7. The minute hand turns 360° in 60 minutes, so in 25 minutes it turns \frac{25}{60} \cdot 360^\circ = 150^\circ
    Write the formula: L = \frac{150}{360} \cdot 2\pi(12)
    Simplify the fraction: L = \frac{5}{12} \cdot 24\pi
    Multiply: L = 10\pi
    Use a calculator: L \approx 31.42 cm
  8. One turn moves the bike one circumference: 2\pi(35) = 70\pi
    Multiply by 3 turns: 3 \cdot 70\pi = 210\pi
    Use a calculator: 210\pi \approx 659.73 cm
  9. A semicircle is 180°: L = \frac{180}{360} \cdot 2\pi(7)
    Simplify the fraction: L = \frac{1}{2} \cdot 14\pi
    Multiply: L = 7\pi
    Use a calculator: L \approx 21.99 ft
  10. Radius 5: L = \frac{60}{360} \cdot 2\pi(5) = \frac{1}{6} \cdot 10\pi = \frac{5\pi}{3} \approx 5.24
    Radius 10: L = \frac{60}{360} \cdot 2\pi(10) = \frac{1}{6} \cdot 20\pi = \frac{10\pi}{3} \approx 10.47
    The arcs have the same measure, but doubling the radius doubles the arc length.

FAQ

What is the formula for arc length in degrees?

Arc length = \frac{\theta}{360^\circ} \cdot 2\pi r, where \theta is the central angle and r is the radius.

What is the difference between arc length and arc measure?

Arc measure is an angle in degrees and equals the central angle. Arc length is a distance, like 12.57 cm, and depends on the radius too.

How do you find arc length without the angle?

You need some way to get the angle. For example, if you know the arc is a fraction of the circle, multiply that fraction by the circumference. If you know the arc’s inscribed angle, double it to get the central angle.

Is arc length the same as circumference?

No. Circumference is the length of the whole circle. Arc length is the length of part of it, so it is a fraction of the circumference.

Summary

  • Arc measure is in degrees. Arc length is a distance.
  • Arc length = \frac{\theta}{360^\circ} \cdot 2\pi r: the fraction of the circle times the circumference.
  • Use the radius, not the diameter, and 360° minus the minor arc for a major arc.
  • Give exact answers in terms of \pi, then round.
  • Arcs with the same measure get longer as the radius grows.

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