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Arc length calculator

The length of a circular arc from the radius and the central angle — arc = r·θ, with θ in radians. Work in either direction: find the arc from radius and angle, the radius from arc and angle, or the angle from arc and radius. It also returns the sector area and the chord length.

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Need Result, Sector area, Chord length? The Arc length calculator derives it from Find, Radius (r), Central angle (θ), Angle unit, Arc length (for reverse modes) in one step. For instance, with Find = Arc length (from radius & angle), Radius (r) = 5, Central angle (θ) = 45, Angle unit = Degrees and Arc length (for reverse modes) = 3.927 it returns Result = 3.927, Sector area = 9.817 and Chord length = 3.827.

How to use it

  1. Enter your values: Find, Radius (r), Central angle (θ), Angle unit, Arc length (for reverse modes).
  2. Read the result instantly: Result, Sector area, Chord length.

Frequently asked questions

How does the Arc length calculator work?

It takes Find, Radius (r), Central angle (θ), Angle unit and Arc length (for reverse modes) and derives Result, Sector area and Chord length from them. The calculation is live as you type, so the result updates on every change.

Which values does the calculator ask for?

5 values: Find, Radius (r), Central angle (θ), Angle unit and Arc length (for reverse modes). Nothing else is required — no account, no file upload.

What does a typical calculation look like?

With Find = Arc length (from radius & angle), Radius (r) = 5, Central angle (θ) = 45, Angle unit = Degrees and Arc length (for reverse modes) = 3.927, the calculator returns Result = 3.927, Sector area = 9.817 and Chord length = 3.827. Those figures come from running this exact tool, so you can reproduce them by entering the same values.

How much does the result change with different inputs?

It moves a lot. Using Find = Radius (from arc & angle), Radius (r) = 10, Central angle (θ) = 90, Angle unit = Radians and Arc length (for reverse modes) = 7.854 instead, Result goes from 3.927 to 0.087 — which is why it is worth testing a few scenarios rather than trusting a single figure.

Which “Find” option should I choose?

You can pick between « Arc length (from radius & angle) », « Radius (from arc & angle) » and « Angle (from arc & radius) ». Each one changes what the calculator works out, so switch and compare — the default is « Arc length (from radius & angle) ».

What does it give for smaller values?

Scaled down to Find = Arc length (from radius & angle), Radius (r) = 3, Central angle (θ) = 23, Angle unit = Degrees and Arc length (for reverse modes) = 1.964, Result comes out at 1.204. The relationship is worth checking at both ends before you rely on a single result.

When would I actually use this?

Checking a derivative or an integral you worked out by hand, finding where a function turns, and getting a numeric answer when no closed form exists.

What is the most common mistake?

Integrating across a discontinuity as if it were not there. A numeric method will happily return a finite value for an integral that diverges — check the domain before trusting the number.

Where do the figures come from, and how current are they?

Symbolic results are exact; numeric ones come from adaptive quadrature or a standard step method, and the tool reports which. Where both are available, compare them — a large gap means the problem is ill-conditioned.

Further reading

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