Area of Sector Calculator 2025: Complete Guide (1000+ Words)
A sector is a pie-shaped portion of a circle. Our 2025 Area of Sector Calculator computes sector area, arc length, chord length, and segment area from radius and central angle with full step-by-step explanations and live visualization.
What is a Sector?
Formed by two radii and an arc. Minor sector: θ ≤ 180°; Major: θ > 180°.
- Radius (r): Distance from center to arc
- Central Angle (θ): Angle at center
- Arc Length (L): Curved edge
- Chord Length (c): Straight line between endpoints
Core Formulas
Sector Area: A = (θ/360) × πr² (degrees)
A = (1/2) r² θ (radians)
Arc Length: L = (θ/360) × 2πr
Chord Length: c = 2r sin(θ/2)
Segment Area: = Sector - Triangle
A = (1/2) r² θ (radians)
Arc Length: L = (θ/360) × 2πr
Chord Length: c = 2r sin(θ/2)
Segment Area: = Sector - Triangle
Example: r=10 cm, θ=60°
A = (60/360) × π × 100 = (1/6) × 314.159 approximately 52.360 cm²
L = (60/360) × 2π × 10 approximately 10.472 cm
c = 2×10×sin(30°) = 10 cm
Triangle = (1/2)×10×10×sin(60°) approximately 43.301 cm²
Segment = 52.360 - 43.301 approximately 9.059 cm²
L = (60/360) × 2π × 10 approximately 10.472 cm
c = 2×10×sin(30°) = 10 cm
Triangle = (1/2)×10×10×sin(60°) approximately 43.301 cm²
Segment = 52.360 - 43.301 approximately 9.059 cm²
Applications
- Engineering: Gears, turbines, cams
- Architecture: Arched windows, domes
- Food: Pizza slices, pie portions
- Astronomy: Field of view, orbits
Step-by-Step: r=8 m, θ=90°
A = (90/360) × π × 64 = 50.265 m²
L = (90/360) × 2π × 8 approximately 12.566 m
Common Mistakes
- Forgetting to convert radians to degrees
- Using diameter instead of radius
- Mixing up sector and segment
Advanced: In Radians
θ in rad: A = (1/2) r² θ
Related Tools
FAQs
Can angle be greater than 360°? No, reduce modulo 360°.
Is sector area always less than full circle? Yes for θ < 360°.
Conclusion
Sectors are fundamental in circular geometry. Our 2025 Area of Sector Calculator delivers instant precision with crystal-clear steps and interactive graphics. Master circular segments today!
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