Circular Sector Area Calculator
Circular sector area calculator: compute area from radius and angle (degrees or radians), plus arc length and perimeter, or solve for radius or angle from area with clear formulas.
Interactive circular sector workspace
Enter the known values, pick the mode, and review the derived sector area, arc length, and perimeter after the calculator finishes.
All modes still report the sector area, arc length, perimeter, and angle conversions.
Internally the calculator uses radians for every formula, converting degrees when needed.
Any length unit (e.g., m, cm, in). Outputs reuse the same unit.
θ must be > 0 and for a single sector typically ≤ 360° (or 2π rad).
Enter in the squared units corresponding to the radius.
Results will appear here once you compute. Area, arc length, perimeter, and angle conversions live in this pane.
Steps and formulas will appear here.
How to use this calculator
Compute the area of a circular sector from radius and angle, or reverse engineer the radius or angle from a known area. The workspace also reports arc length and sector perimeter to help with geometry homework, exam prep, or engineering checks.
Designed for geometry, engineering, and exam prep
Handles degree/radian angles, enforces positive inputs, and exposes the formulas textbooks and standards references cite.
Author: CalcDomain Math Team
Reviewed by: Geometry specialist
Last updated: 2025
This calculator is provided for educational and general engineering support. For high-stakes design work, always verify results with your organization’s approved methods and codes.
What is a circular sector?
A circular sector is the region of a circle bounded by two radii and the arc between them – like a “slice of pizza.” It is completely determined by the radius r and the central angle θ.
Formulas: area, arc length, and perimeter
If the angle θ is measured in radians, the key formulas are:
s = r θ
P = 2 r + s = 2 r + r θ
where:
- A is the sector area,
- s is the arc length, and
- P is the sector perimeter (two radii plus the arc).
If the angle is given in degrees θdeg, convert to radians first:
Rearranging the formulas
- For angle (given A and r): θ = 2 A / r².
- For radius (given A and θ): r = √(2 A / θ).
The calculator automates these rearrangements and handles unit conversions for you.
Worked example
Suppose r = 5 m and θ = 60°.
- Convert to radians: θ = 60° · π / 180 = π / 3.
- Area: A = ½ · 25 · (π / 3) = 25π / 6 ≈ 13.09 m².
- Arc length: s = r θ = 5 · (π / 3) ≈ 5.24 m.
- Perimeter: P = 2 r + s = 10 + 5.24 ≈ 15.24 m.
FAQ: circular sector area calculator
What angle range is valid for a sector?
For a standard sector, the central angle satisfies 0 < θ ≤ 2π radians (or 0° < θ ≤ 360°). Angles larger than a full turn are treated as multiple turns instead of one sector.
Which units should I use for radius?
Use any consistent length unit. The sector area appears in the corresponding squared unit (e.g., m² if r is meters) while arc length and perimeter use the radius unit.
Why do engineers often prefer radians?
Many formulas simplify when angles are in radians. For sectors, the area is A = ½ r² θ and the arc length is s = r θ without extra conversion factors, reducing mistakes and making calculus easier.
Can this calculator replace formal design checks?
No. This tool uses standard formulas and double-precision arithmetic, which is adequate for most classroom and practical calculations. For safety-critical or regulated design work, follow the applicable codes, safety factors, and internal verification procedures.
Full original guide (expanded)
The following sections preserve the original supporting materials that reference related tools and practical tips.
Related Core Math & Geometry tools
Practical tips
- Keep angle units consistent and avoid mixing degrees and radians in the same formula.
- For very small angles, area and arc length scale linearly with θ; doubling θ doubles both.
- When checking drawings, compare sector area with the full circle area A_circle = π r².
- In CAD or FEA software, cross-check one manual sector calculation against the tool’s output.