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Calculator Answers

Arc Length Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

An arc length calculator is a geometry utility that computes the distance along a circular arc. This one solves the relationship arc length equals radius multiplied by the central angle in radians, in any direction: give it two of arc length, radius and angle and it returns the third, in degrees or radians, with an optional sector area. It does not perform calculus arc length, the definite integral of the square root of one plus the derivative squared, which is the method for a general curve rather than a circular one. Students and makers use it to find the distance along a curve of known radius.

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.

See also: Circle Calculator – Area & Circumference and Block Circle Generator for Minecraft.

Quick Answer

Calculate the arc length of a circular arc. Enter any two of radius, central angle and arc length to find the third, in degrees or radians, with the sector area alongside it.

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

Arc length s

5.235988

θ = 1.047198 rad; sector area = 13.089969

Radius5
Arc length5.235988
Central angle (rad)1.047198
Sector area13.089969
Formulas = r · θ = 5 · 1.047198 rad = 5.235988

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.

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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.

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Circle Calculator – Area & CircumferenceCalculate the area of a circle from the radius, diameter, circumference, or area itself. Shows the circle area formula A = πr² with your own numbers substituted, and names the exact multiple of π whenever the coefficient is a clean one.Block Circle Generator for MinecraftGenerate a block circle of any whole diameter up to 128 blocks, as an outline or filled in. Shows the grid, the block count and the exact rule used to pick the blocks.

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.