Sector Area
A sector is a 'pie slice' of a circle—bounded by two radii and the arc between them. Sector area = (θ/360°) × πr²; arc length = rθ (radians). Used in pizza slices, pie charts, and windshield wipers.
Why This Mathematical Concept Matters
Why: Sectors appear in pizza slices (45°), pie charts (proportional angles), windshield wipers, radar sweeps, and sprinkler coverage. Sector area is a fraction of the full circle.
How: Use A = (θ/360°) × πr² for degrees, or A = ½r²θ for radians. Arc length L = rθ (radians). Chord c = 2r sin(θ/2). Segment = sector minus triangle.
- ●A standard pizza slice is typically 45° (1/8 of a circle).
- ●Pie charts use sectors—25% of data = 90° angle.
- ●Sector perimeter = 2r + arc length (includes both radii).
Sector Area, Arc Length & Segment Calculator
Enter radius and central angle (degrees or radians) to get sector area, arc length, chord length, segment area, and sector perimeter.
◔ Real-World Sector Examples — Click to Load
Input Dimensions
Sector Properties Radar
Property Comparison
Sector vs Full Circle
Step-by-Step Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
Sector area = (θ/360°) × πr² — a fraction of the full circle.
— Formula
Chord length = 2r sin(θ/2). For 180°, chord = diameter.
— Geometry
Key Takeaways
- Sector area = (θ/360°) × πr² — a fraction of the full circle
- Arc length = r × θ (radians) — the curved edge of the sector
- Chord length = 2r × sin(θ/2) — the straight line between arc endpoints
- Segment area = sector area − triangle area — the "cap" between arc and chord
- Real-world sectors: pizza slices, pie charts, windshield wipers, radar sweeps, sprinkler coverage
Did You Know?
How Sector Calculations Work
A circle sector is a "pie slice" bounded by two radii and the arc between them. All properties derive from the radius (r) and central angle (θ).
Sector Area
A = (θ/360°) × πr². The sector is a fraction of the full circle — if θ = 90°, you get 1/4 of the circle area. In radians: A = ½r²θ.
Arc Length
L = r × θ (radians). The arc is the curved edge. In degrees: L = (θ/360°) × 2πr. The arc length grows linearly with the angle.
Segment vs Sector
The segment is the region between the arc and the chord. Segment area = sector area − triangle area (formed by the two radii and chord). The chord length is c = 2r × sin(θ/2).
Expert Tips for Sector Problems
Use Radians for Calculus
In calculus, arc length and sector area formulas are simpler in radians: L = rθ, A = ½r²θ. Convert degrees to radians: θ_rad = θ_deg × π/180.
Sector vs Segment
Sector includes the center; segment does not. For θ < 180°, the segment is the "cap" above the chord. For θ = 180°, segment area = semicircle − triangle.
Sector Perimeter
Sector perimeter = 2r + arc length. It includes both radii and the curved arc. Not to be confused with the full circle circumference.
Special Angles
90° = quarter circle; 180° = semicircle; 360° = full circle. For 90°, sector area = πr²/4 and arc length = πr/2.
Sector vs Full Circle Comparison
| Property | Sector (θ) | Full Circle | Relationship |
|---|---|---|---|
| Area | (θ/360°) × πr² | πr² | Fraction θ/360° |
| Arc Length | r × θ (rad) | 2πr | Fraction θ/(2π) |
| Chord | 2r × sin(θ/2) | 0 (degenerate) | N/A |
| Perimeter | 2r + arc | 2πr | Different (includes radii) |
Frequently Asked Questions
What is the area of a sector?
A = (θ/360°) × πr², where θ is the central angle in degrees. In radians: A = ½r²θ. The sector is a fraction of the full circle area.
What is the difference between sector and segment?
A sector is bounded by two radii and the arc (like a pizza slice). A segment is bounded by the arc and the chord — it does not include the center. Segment area = sector area − triangle area.
How do I convert degrees to radians?
Multiply by π/180. So 90° = π/2 rad, 180° = π rad, 360° = 2π rad. To convert radians to degrees, multiply by 180/π.
What is the arc length formula?
Arc length = r × θ (when θ is in radians). In degrees: L = (θ/360°) × 2πr. The arc is the curved portion of the sector.
What is the chord length?
Chord length = 2r × sin(θ/2). It is the straight line connecting the two endpoints of the arc. For θ = 180°, chord = 2r (the diameter).
What are real-world sector examples?
Pizza slices, pie charts, windshield wipers, radar sweeps, sprinkler coverage, clock hands, protractor arcs, and any circular "slice" shape.
Sector by the Numbers
Official & Trusted Sources
Disclaimer: This calculator provides mathematically precise results based on standard geometric formulas. Results are limited by floating-point precision (~15 significant digits). For critical engineering or architectural applications, verify with domain-specific tools. Not a substitute for professional analysis.