Arc Length Calculator
Calculate arc length, sector area, and chord length from a circle's radius and central angle. Free, instant results.
How to Use the Arc Length Calculator
- Enter the radius (r) of the circle.
- Enter the central angle (θ) in radians.
- View the arc length, sector area, and chord length instantly.
Features of the Arc Length Calculator
- Calculates arc length, sector area, and chord length from the same two inputs.
- Works with any central angle in radians, including angles greater than a full rotation.
- Live results as you type, with no separate calculate step required.
- Free to use with no sign-up required.
Arc Length, Sector Area, and Chord Length Formulas
For a circle of radius r and a central angle θ measured in radians, three related quantities can be found from the same two values.
Arc Length: L = r × θ
Sector Area: A = ½ × r² × θ
Chord Length: C = 2 × r × sin(θ ÷ 2)
Arc Length Reference Table
| Radius (r) | Angle (θ, radians) | Arc Length | Sector Area | Chord Length |
|---|---|---|---|---|
| 5 | 1.2 | 6 | 15 | 5.6464 |
| 10 | π/2 ≈ 1.5708 | 15.708 | 78.5398 | 14.1421 |
| 3 | 2 | 6 | 9 | 5.0488 |
| 7 | π ≈ 3.1416 | 21.9911 | 76.969 | 14 |
| 4 | π/3 ≈ 1.0472 | 4.1888 | 8.3776 | 4 |
How to Calculate Arc Length
To find the arc length, multiply the radius by the central angle in radians.
Suppose you have a circle with a radius of 5 and a central angle of 1.2 radians. Arc Length = 5 × 1.2 = 6.
Thus, the arc length is 6.
Solved Examples on Arc Length, Sector Area, and Chord Length
1. A Quarter-Circle Sector
Example: Find the arc length and sector area for r = 10 and θ = π/2.
Arc Length = 10 × (π/2) ≈ 15.708. Sector Area = ½ × 10² × (π/2) ≈ 78.5398.
2. A Semicircle
Example: Find the arc length and chord length for r = 7 and θ = π.
Arc Length = 7 × π ≈ 21.9911. Chord Length = 2 × 7 × sin(π/2) = 14, which is the diameter, as expected for a semicircle.
3. Chord Length for a Small Angle
Example: Find the chord length for r = 3 and θ = 2.
Chord Length = 2 × 3 × sin(1) ≈ 5.0488.
4. A 60° Sector
Example: Find the arc length, sector area, and chord length for r = 4 and θ = π/3 (60°).
Arc Length ≈ 4.1888. Sector Area ≈ 8.3776. Chord Length = 2 × 4 × sin(π/6) = 4.
1. Why does the formula use radians instead of degrees?
The arc length formula L = r × θ only works directly when θ is measured in radians, since radians express angle as a ratio of arc length to radius. To use degrees, first convert with θ(radians) = θ(degrees) × π ÷ 180.
2. What’s the difference between arc length and chord length?
Arc length measures the distance along the curved edge of the circle between two points, while chord length measures the straight-line distance connecting those same two points. Arc length is always greater than or equal to chord length.
3. What is a sector of a circle?
A sector is the pie-slice-shaped region of a circle bounded by two radii and the arc between them. Its area is found using the sector area formula A = ½ × r² × θ.
4. Can the central angle be larger than a full circle?
Yes, mathematically θ can exceed 2π radians (360°), representing more than one full rotation. In that case, arc length and sector area keep increasing, while chord length repeats periodically since it depends on sin(θ/2).
5. What happens when the central angle is zero?
When θ = 0, the two radii overlap, so arc length, sector area, and chord length are all 0.
6. Where are these formulas used in practice?
These formulas are used in engineering for designing curved components, in navigation for calculating distances along circular paths, and in geometry problems involving pie charts, gears, and circular arcs.