Education
Arc Length Calculator
Pick a mode (solve for arc length, central angle, or radius). Enter the known values; the calculator returns the missing one along with the sector area. Supports degrees and radians.
Solve for
Unit
Notes
- The formula s = r · θ assumes the central angle is in radians. Degree inputs are converted internally.
- Sector area is the slice cut out by the arc, computed as ½ · r² · θ (with θ in radians).
- Use the unit circle calculator if you only need sin/cos/tan of a common angle.
Arc length s
5.235988
θ = 1.047198 rad; sector area = 13.089969
A full circle has central angle 2π rad and arc length 2πr (the circumference). The arc length formula s = r · θ is the cleanest way to remember the relationship.
Examples
r = 5, θ = 60°
s ≈ 5.236; sector area ≈ 13.09
r = 10, θ = π/2 rad
s ≈ 15.708; sector area ≈ 78.54
s = 7, r = 5
θ = 1.4 rad ≈ 80.21°
s = 20, θ = 2 rad
r = 10
How it works
The arc length formula has three quantities. Any two of them determine the third. The calculator handles the unit conversion for degrees and radians automatically.
Arc length · s = r · θ (θ in radians)
Sector area · A = ½ · r² · θ
Degree to radian: multiply by π / 180. Radian to degree: multiply by 180 / π.
Related calculators
- Circle calculator for radius, diameter, circumference, and full-circle area.
- Unit circle calculator for sin, cos, tan, and the unit-circle coordinate of an angle.
- Angle converter for converting between degrees, radians, gradians, turns, and arcseconds.
- Scientific calculator for general arithmetic checks.
- All education calculators.
Frequently asked questions
s = r · θ, where s is the arc length, r is the radius of the circle, and θ is the central angle in radians. If your angle is in degrees, multiply by π / 180 first.
The clean formula s = r · θ only works when θ is dimensionless. A radian is defined as the angle whose arc length equals the radius, so the formula falls out of the definition. Degrees are a different unit and need conversion.
Arc length is the length of the curved edge of a slice; sector area is the area of the slice (radius x radius x angle / 2 with angle in radians). The calculator shows both.
A full circle has arc length equal to the circumference, 2πr. The maximum sensible central angle is 2π radians (360°); beyond that you wrap around the circle.
Yes. Pick a different mode in the calculator and enter the two known values. Solving for the radius requires a nonzero central angle.
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