Arc Length Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Arc Length.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The distance along a portion of a circle's circumference, determined by the central angle and the radius.

Imagine walking along a circular track but only covering a portion of the full loop. The arc length is how far you actually walked. If you walk a quarter of the circle ( $90°$ ), you cover a quarter of the circumference. The fraction of the full circle you cover determines the fraction of the circumference you walk.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Arc length is the part of the circle's perimeter you cover, scaled by what fraction of $360°$ the central angle is.

Common stuck point: The procedure for arc length is the easy part; the trap is using $\pi r^2$ (area) instead of $2\pi r$ (circumference) for the whole. Asking "Am I asked for a length along the circle's edge (not an angle and not an enclosed area)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I asked for a length along the circle's edge (not an angle and not an enclosed area)?

Worked Examples

Example 1

easy

A circle has radius

6

cm. Find the arc length intercepted by a central angle of

60°

Arc length intercepted by a 60° central angle in a circle of radius 6 cm

Answer

s = 2\pi \approx 6.28

First step

Step 1: Write the arc length formula using degrees:

s = \frac{\theta}{360°} \times 2\pi r

Full solution

2
Step 2: Substitute $\theta = 60°$ and $r = 6$ cm: $s = \frac{60}{360} \times 2\pi(6)$ .
3
Step 3: Simplify the fraction: $\frac{60}{360} = \frac{1}{6}$ , so $s = \frac{1}{6} \times 12\pi$ .
4
Step 4: Compute: $s = 2\pi \approx 6.28$ cm.

Arc length is the fraction of the full circumference determined by the central angle. A 60° angle is one-sixth of 360°, so the arc is one-sixth of the full circumference

2\pi(6) = 12\pi

, giving

2\pi

cm.

Example 2

medium

A circle has radius

5

m. Find the arc length subtended by a central angle of

\frac{3\pi}{4}

radians.

Arc length for a central angle of 3π/4 radians in a circle of radius 5 m

Example 3

medium

A circle has radius

10

cm. Find the arc length for a central angle of

135°

in terms of

\pi

Example 4

medium

A bicycle wheel has radius

35

cm. How far does it travel in one full revolution?

Example 5

hard

An arc length of

12\pi

on a circle has central angle

\dfrac{4\pi}{5}

radians. Find the radius.

Example 6

hard

A circle has diameter

20

. Find the arc length of a

108°

arc in terms of

\pi

Example 7

challenge

A circular running track has inner radius

36.5

m. An athlete runs

400

m along the inner edge. Through how many radians has the athlete moved? Round to two decimals.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A circle has a radius of

10

cm. What is the arc length for a central angle of

90°

Arc length for a 90° central angle in a circle of radius 10 cm

Example 2

hard

A pendulum of length

80

cm swings through an angle of

0.4

radians. How far does the tip of the pendulum travel in one complete swing (from one side to the other and back)?

Example 3

easy

A circle has radius

4

. Find the arc length for a central angle of

2

radians.

Example 4

easy

A circle has radius

3

. Find the arc length for a central angle of

\pi

radians.

Example 5

easy

A circle has circumference

24\pi

. Find the arc length of a

90°

arc.

Example 6

medium

Find the arc length on a circle of radius

15

subtended by a central angle of

\dfrac{2\pi}{5}

radians.

Example 7

medium

An arc of length

4\pi

is on a circle of radius

8

. Find the central angle in radians.

Example 8

medium

An arc has length

10

m on a circle of radius

4

m. Find the central angle in degrees (to the nearest degree).

Example 9

medium

A clock's minute hand is

10

cm long. How far does its tip travel in 15 minutes?

Example 10

medium

An arc length of

7\pi

corresponds to a central angle of

140°

. Find the radius.

Example 11

medium

A circle has radius

6

. Find the arc length of a

240°

arc in terms of

\pi

Example 12

hard

A car wheel of radius

30

cm rotates at

200

rpm. Find the linear speed of a point on the rim in meters per minute.

Example 13

hard

A racetrack has two straight

100

m sections connected by two semicircular ends of radius

30

m. What is the total length of one lap?

Example 14

hard

A circle of radius

r

has a

60°

sector. The arc length of that sector is

2\pi

. Find

r

Example 15

hard

Two pulleys with radii

4

and

9

rotate so the linear speed of their belts is the same. If the larger pulley rotates at

\omega_2 = 4

rad/s, find

\omega_1

(the smaller pulley's angular speed).

Example 16

hard

A windshield wiper

50

cm long sweeps an angle of

120°

. Find the area swept and the arc length traced by the tip. Give arc length in terms of

\pi

Example 17

challenge

Earth's radius is approximately

6371

km. A

1°

change in latitude corresponds to how many kilometers of arc length along a meridian? Round to the nearest km.

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Related Concepts

Circumference Central Angle Sector Area Radians

Background Knowledge

These ideas may be useful before you work through the harder examples.

circumferencecentral angle