s=r\thetaVariables
- s: arc length
- r: radius
- θ: central angle in radians
How to use this formula
Calculates an arc length from a circle radius and a central angle measured in radians.
Important notes
- Convert degrees to radians before using this form.
- For a full circle, θ = 2π and s = 2πr.
Quick example
With r = 3 and θ = π/2, the arc length is 3π/2.
Applicability, worked calculation, and verification
Domain and applicability
Apply Circular Arc Length only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- Convert degrees to radians before using this form.
- Use one consistent unit system and confirm whether lengths, areas, volumes, and angles use the dimensions implied by the formula.
Do not use this formula when
- Do not use Circular Arc Length when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units.
Boundary and special cases
- For Circular Arc Length, check zero, negative, and extreme input values before relying on the result.
- When using Circular Arc Length, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form s=r\theta for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For r=3 and θ=π/2 radians, s=rθ=3π/2≈4.712.
- Convert the central angle to radians if necessary.
- Multiply radius 3 by π/2.
- Report the result in the same length unit as the radius.
Independent verification
A quarter of a circle with circumference 6π has length 6π/4=3π/2.
Common mistakes
- Before substituting values into Circular Arc Length, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- For the Circular Arc Length, do not report a length, area, or volume with the wrong unit dimension, and keep intermediate precision until the final rounding step.
Continue the workflow
Use Circular Arc Length in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
s=r\theta. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Circular Arc Length.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Passed the core formula indexing gate.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Circular Arc Length used for?
Calculates an arc length from a circle radius and a central angle measured in radians.
Can I copy this formula as LaTeX?
Yes. Copy s=r\theta or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.