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 a fraction of the circumference, where the fraction equals the ratio of the central angle to the full angle (360° or 2\pi).

Common stuck point: Make sure the angle and formula match—use radians with s = r\theta and degrees with s = \frac{\theta}{360} \cdot 2\pi r.

Sense of Study hint: When you see an arc length problem, first identify the radius and central angle. Then check whether the angle is in degrees or radians. Use s = r\theta for radians, or s = \frac{\theta}{360} \cdot 2\pi r for degrees. Finally, substitute and simplify.

Worked Examples

Example 1

easy
A circle has radius 6 cm. Find the arc length intercepted by a central angle of 60°.

Solution

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

Answer

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.

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°?

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)?
← Back to Arc Length Formula Explained Practice Mode

Background Knowledge

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

circumferencecentral angle