Arc Length Calculator
Calculate the arc length, sector area, and chord length from the radius and central angle of a circle. See also Area of Sector Calculator and Area of Circle Calculator.
How to Calculate Arc Length
Arc length is the distance along the curved part of a circle between two points. To calculate it, you need the radius and the central angle subtended by the arc. If the angle is in radians, simply multiply: L = rθ. If in degrees, convert first or use L = (θ/360) × 2πr. The calculator also computes the sector area (the "pie slice" enclosed by the arc) and the chord length (the straight-line distance between the arc's endpoints).
Arc Length Formula
L = r × θ (θ in radians)
L = (θ/360) × 2πr (θ in degrees)
Sector Area = ½ × r² × θ (radians)
Chord Length = 2r × sin(θ/2)
Example
Find the arc length for radius 10 and angle 90°:
L = (90/360) × 2π × 10
L = 0.25 × 20π
L = 5π
L ≈ 15.7080 units
Sector Area = ½ × 100 × (π/2) ≈ 78.5398
Chord Length = 2 × 10 × sin(45°) ≈ 14.1421
Arc Length Reference Table
| Radius | Angle (°) | Arc Length | Sector Area | Chord Length |
|---|---|---|---|---|
| 5 | 30° | 2.6180 | 6.5450 | 2.5882 |
| 5 | 60° | 5.2360 | 13.0900 | 5.0000 |
| 5 | 90° | 7.8540 | 19.6350 | 7.0711 |
| 5 | 180° | 15.7080 | 39.2699 | 10.0000 |
| 10 | 30° | 5.2360 | 26.1799 | 5.1764 |
| 10 | 45° | 7.8540 | 39.2699 | 7.6537 |
| 10 | 60° | 10.4720 | 52.3599 | 10.0000 |
| 10 | 90° | 15.7080 | 78.5398 | 14.1421 |
| 10 | 120° | 20.9440 | 104.7198 | 17.3205 |
| 10 | 180° | 31.4159 | 157.0796 | 20.0000 |
| 10 | 270° | 47.1239 | 235.6194 | 14.1421 |
| 10 | 360° | 62.8319 | 314.1593 | 0.0000 |
When to Use the Arc Length Formula
Use the arc length formula whenever you need to find the distance along a curved path rather than a straight line. Common situations include:
- Curved fence sections — when fencing follows a circular boundary (around a roundabout or curved garden edge), arc length determines how much fencing to buy
- Road curves and highway bends — engineers calculate arc length to determine road distance around curves, paint line requirements, and guard rail lengths
- Belt and pulley systems — the length of belt wrapping around a pulley follows an arc, critical for sizing belts correctly
- Satellite dish curvature — the curved surface follows an arc, and the arc length helps in manufacturing the dish surface
- Running track curves — the curved portions of an athletic track are arcs; lane stagger is calculated from arc length differences
- Clock hand distance — how far the tip of a clock hand travels in a given time is an arc length problem
The formula L = rθ (radians) gives the exact distance along the curve. This is always longer than the chord (straight-line) distance between the same two points.
Solved Examples
Example 1: Curved Garden Fence
A garden has a curved edge that follows a 6-meter radius circle through a 90° arc. How much fencing material do you need for the curved section?
r = 6 m, θ = 90°
Convert: 90° × (π/180) = π/2 radians
L = r × θ
L = 6 × (π/2)
L = 6 × 1.5708
L ≈ 9.42 meters of curved fencing
Example 2: Distance Traveled by a Clock's Minute Hand
A clock's minute hand is 12 cm long. How far does the tip travel in 20 minutes?
r = 12 cm
20 minutes = 20/60 of full rotation = 1/3 of 360° = 120°
θ = 120° × (π/180) = 2π/3 radians
L = 12 × (2π/3)
L = 12 × 2.0944
L ≈ 25.13 cm traveled by the tip
Example 3: Running Track Curve
A running track has semicircular ends with an inner radius of 36.5 m. What is the distance around one curved end (inner lane)?
r = 36.5 m, θ = 180° = π radians
L = r × θ
L = 36.5 × π
L = 36.5 × 3.14159
L ≈ 114.67 meters per curve
Example 4: Highway Curve Road Marking
A highway curves with a radius of 200 meters through a 45° angle. How long is the road center line along this curve?
r = 200 m, θ = 45°
θ = 45° × (π/180) = π/4 = 0.7854 radians
L = 200 × 0.7854
L ≈ 157.08 meters of center line
Common Mistakes When Calculating Arc Length
Using degrees directly in L = rθ
The formula L = rθ only works when θ is in radians. If θ = 90°, you must convert: θ = 90 × π/180 = π/2 ≈ 1.5708. Using 90 directly gives a wildly wrong answer.
Confusing arc length with chord length
Arc length follows the curve; chord length is the straight line between endpoints. Arc is always longer. For a 90° arc with r=10: arc = 15.71, chord = 14.14.
Forgetting that full circle = 2π radians (not π)
A full circle is 360° = 2π radians. The circumference (full arc) is 2πr, not πr. A semicircle (180°) gives arc = πr.
Key Takeaways
- Arc length formula: L = rθ (θ in radians) or L = (θ/360) × 2πr (θ in degrees).
- Arc length is always greater than chord length (except when angle = 0, where both are 0).
- The full circumference is a special case: L = 2πr when θ = 360° (2π radians).
- To convert degrees to radians: multiply by π/180. To go back: multiply by 180/π.
- Arc length is proportional to the angle — double the angle means double the arc length (for fixed radius).
- In running track design, each lane adds about 2π × lane_width ≈ 6.28 m per semicircular curve, which is why lanes are staggered.
Frequently Asked Questions
What is arc length?
Arc length is the distance measured along the curved line of a circle (or any curve) between two points. For a circle, it depends on the radius and the central angle.
What is the difference between arc length and chord length?
Arc length follows the curve of the circle, while chord length is the straight-line distance between the two endpoints. Arc length is always greater than or equal to chord length (equal only when the angle is 0).
What is the arc length of a full circle?
The arc length of a full circle (360° or 2π radians) is the circumference: L = 2πr. For example, a circle with radius 10 has circumference ≈ 62.832.
How do I find the angle if I know the arc length?
Rearrange the formula: θ = L/r (in radians) or θ = (L × 360)/(2πr) in degrees. For example, if L = 15.708 and r = 10, then θ = 15.708/10 ≈ 1.5708 rad = 90°.
Can arc length be negative?
No. Arc length is always a positive value representing a physical distance. Both the radius and angle must be positive for a meaningful result.