Key takeaways
- The arc length of a circle is L = (θ / 360) × 2 × π × r.
- θ (theta) is the central angle in degrees, out of 360.
- A 90° arc is always exactly one-quarter of the full circumference.
- In radians, the formula simplifies to L = r × θ.
In this guide
What is arc length?
An arc is a curved section of a circle's outer edge, like one bend of a running track or a slice of pie crust. Arc length is how long that curved section is, measured in regular units, not square units.
Why does arc length matter?
It tells you the actual curved distance along part of a circle, which straight-line measurements can't capture.
A 90° arc is always exactly one-quarter of the full circumference, no matter the radius, since 90 ÷ 360 = 0.25 every time — the same fraction rule that applies to sector area.The formula
| Symbol | What it means | Example |
|---|---|---|
| L | The arc length you want to find | 15.71 m |
| θ (theta) | The central angle the arc spans, in degrees | 60° |
| r | The radius of the full circle | 15 m |
How to find the arc length step by step
- 1Measure the radius of the full circle.
- 2Measure or find the central angle the arc spans, in degrees.
- 3Divide the angle by 360. This gives the fraction of the full circle the arc represents.
- 4Find the full circle's circumference.
2 × π × r. - 5Multiply that fraction by the full circumference.
Now try it yourself. Change the radius and angle in the calculator below and watch the arc length update, with every step shown.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Confusing arc length with [sector area](/calculator/area-sector). Arc length is just the curved edge, a distance; sector area is the whole pie-slice-shaped region.
- Forgetting to divide by 360. Skipping this step gives you the full circumference, not just part of it.
- Mixing up degrees and radians in the same calculation. Always double check which unit your angle is in first.
Tips and tricks
- Convert between degrees and radians with the degrees to radians calculator whenever your angle is in the wrong unit.
- For the full distance around the circle instead of just part of it, use the circumference calculator.
Where you'll use it in real life
- Sports: measuring the curved section of a running track.
- Cooking: finding the crust length of a pizza slice.
- Architecture: working out material needed for a curved architectural feature.
- Landscaping: measuring a curved garden path or border.
Quick summary
To find arc length, divide the central angle by 360 to get a fraction of the full circle, then multiply by the full circumference (2πr). A 90° arc is always exactly a quarter of the full circumference. Use the calculator above to check your own arc instantly.
Worked example
A curved section of a running track has a radius of 15 meters and spans a 60° turn. How long is that curved section of track?
- 01L = (θ / 360) × 2πr
- 02L = (60 / 360) × 2π(15)
- 03L = 0.1667 × 94.25
- 04L = 15.71 m
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I find arc length if the angle is in radians?
If the angle is already in radians, arc length is simply L = r × θ (radius times angle), no fraction needed since radians measure the angle directly in terms of the radius.
What's the difference between arc length and sector area?
Arc length is a distance, the length of the curved edge alone. [Sector area](/calculator/area-sector) is the area of the whole pie-slice-shaped region bounded by that arc and two radius lines.
What if the angle is 360°?
Then the arc is the entire circle, and the formula simplifies to the regular [circumference formula](/calculator/circumference-circle), 2πr.
Does a bigger radius always mean a longer arc for the same angle?
Yes. For the same central angle, a larger circle has a longer arc, since arc length is directly proportional to the radius.
