Sector Area Calculator

Sector Area Calculator — Find Area of a Circular Sector

Calculate the area of a circular sector using radius and central angle. Learn formulas for degrees and radians. For related tools, see our circle area calculator.

Table of Contents

1. 1What Is a Circular Sector?
2. 2Area Formula (Degrees)
3. 3Area Formula (Radians)
4. 4Step-by-Step Examples
5. 5More Sector Examples
6. 6Arc Length of a Sector
7. 7Common Sector Fractions
8. 8Sector vs. Segment
9. 9Common Mistakes
10. 10Real-World Applications
11. 11Practice Problems
12. 12Tips for Mastery
13. 13Final Thoughts
14. 14FAQ: Common Questions

Calculate Sector Area

Enter radius and central angle. Choose degrees or radians. Click Calculate.

Radius (r)

Central Angle (θ)

Angle Unit Degrees (°)Radians (rad)

Sector area will appear here.

Note: A sector is a "pizza slice" of a circle. Degrees: A = (θ/360°)πr². Radians: A = ½r²θ.

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Circular Sector – Complete Explanation

A circular sector is a "pizza slice" of a circle — the region between two radii and an arc.

Sector Area Formulas

Degrees: A = (θ/360°) × πr²
Radians: A = ½r²θ
Arc: L = (θ/360°) × 2πr (degrees) or L = rθ (radians)

1. What Is a Circular Sector?

The region bounded by two radii and the connecting arc.

2. Area Formula (Degrees)

A = (θ/360°) × πr²

3. Area Formula (Radians)

A = ½r²θ

The radian formula is simpler and preferred in calculus and physics.

4. Step-by-Step Examples

90° Sector (r=10)

A = (90/360)×π×100 = 25π ≈ 78.540

r=6, θ=1.5 rad

A = ½×36×1.5 = 27

5. More Sector Examples

r=4, θ=180°: 8π ≈ 25.133 (semicircle)

r=3, θ=120°: 3π ≈ 9.425

r=8, θ=45°: 8π ≈ 25.133

r=5, θ=π/2: 25π/4 ≈ 19.635

6. Arc Length of a Sector

L = (θ/360°) × 2πr = rθ (radians)

7. Common Sector Fractions

90° = 1/4 | 60° = 1/6 | 45° = 1/8 | 30° = 1/12

8. Sector vs. Segment

A segment is a sector minus the triangular part: A_segment = ½r²(θ - sinθ).

9. Common Mistakes

• Using degrees in A = ½r²θ.
• Forgetting to square radius.
• Using diameter instead of radius.
• Confusing sector with segment.

10. Real-World Applications

• Pizza sizing.
• Irrigation coverage.
• Pie charts.
• Radar/sonar scanning.
• Stage lighting.

11. Practice Problems

P1: r=6, θ=60° → 6π ≈ 18.850

P2: r=5, θ=2 rad → 25

P3: A=12π, r=6 → θ=120°

P4: r=8, θ=90° → arc = 4π ≈ 12.566

12. Tips for Mastery

• Sector = fraction of full circle.
• Use radians for A = ½r²θ.
• Use degrees for A = (θ/360°)πr².
• Memorize common fractions.
• Sector Area = ½ × r × Arc Length.

13. Final Thoughts

The circular sector connects proportional reasoning to circle geometry. Master both formulas thoroughly.

14. Frequently Asked Questions

A = (θ/360°) × πr².

A = ½r²θ.

Degrees: L = (θ/360°)×2πr. Radians: L = rθ.

A sector includes the triangular area to the center. A segment is just the cap outside the chord.

Yes — it would wrap around the circle more than once.

The formula A = ½r²θ is simpler and connects to calculus.

Sector Area = ½ × r × Arc Length.

Explore Math Topics

Continue your learning journey with our other math guides

🔢Basic Arithmetic 📐Pre-Algebra 📊Algebra 📏Geometry 📈Statistics

Related Calculators

Explore these other geometry tools:

• 🔹 Circle Area Calculator — Full circle area.
• 🔹 Circle Circumference Calculator — Circumference.
• 🔹 Arc Length Calculator — Arc length only.
• 🔹 Area Calculator — Area of common shapes.
• 🔹 Perimeter Calculator — Distance around shapes.

Conclusion

The circular sector connects proportional reasoning to circle geometry. Master both formulas thoroughly.

Readers who wish to continue practicing can explore the companion learning resource for guided exercises.