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 |
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.