A=\frac12 r^2\thetaVariables
- A: sector area
- r: circle radius
- θ: central angle in radians
How to use this formula
Calculates sector area when the central angle is measured in radians.
Important notes
- For degrees, use A=(θ/360°)πr².
- The radius and area units must be compatible.
Quick example
With r=3 and θ=π/2, A=9π/4.
Applicability, worked calculation, and verification
Domain and applicability
Apply Area of a Circular Sector only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- For degrees, use A=(θ/360°)πr².
- For the Area of a Circular Sector, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- 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 Area of a Circular Sector when the stated lengths, angles, or geometric conditions do not match the figure or use inconsistent units.
Boundary and special cases
- For Area of a Circular Sector, check zero, negative, and extreme input values before relying on the result.
- When using Area of a Circular Sector, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form A=\frac12 r^2\theta for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For r=3 and θ=π/2 radians, A=½·9·π/2=9π/4≈7.069.
- Use radians for θ.
- Square the radius to obtain 9.
- Multiply by θ/2.
Independent verification
A 90° sector is one quarter of the circle area 9π, which is 9π/4.
Common mistakes
- When copying Area of a Circular Sector, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Area of a Circular Sector, 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 Area of a Circular Sector 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
A=\frac12 r^2\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 Area of a Circular Sector.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Kept noindex until the missing evidence is supplied.
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 Area of a Circular Sector used for?
Calculates sector area when the central angle is measured in radians.
Can I copy this formula as LaTeX?
Yes. Copy A=\frac12 r^2\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.