Education
Arc Length Calculator
Last updated: June 19, 2026
An arc length calculator is a geometry and calculus utility that computes the distance along a curved line or circular arc. The calculator supports circular arc calculations using the formula arc length equals radius multiplied by the central angle in radians, and calculus-based curve calculations using definite integration of the square root of one plus the derivative squared. It outputs exact analytical answers and decimal approximations. Physics and math students use this tool to determine travel distance along curved trajectories.
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.
Quick Answer
Calculate the arc length of a circle or a mathematical function. Enter the radius and central angle to find the precise curve distance.
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
- Arc Length Formula
- 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.
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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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