Sector Area Calculator: Calculate Sector Area, Arc Length & Angle
A sector area calculator computes the area, arc length, perimeter, and central angle of a circular sector (pie-slice shaped region) using geometric formulas, where sector area equals half the radius squared times the central angle in radians (A = ½r²θ) or the fraction of the circle's area based on the angle (A = (θ/360°) × πr²), arc length equals radius times angle in radians (L = rθ) or the fraction of circumference (L = (θ/360°) × 2πr), and sector perimeter equals arc length plus two radii (P = L + 2r). This comprehensive geometric tool performs calculations including finding sector area from radius and angle, calculating arc length, determining central angle from area, computing sector perimeter, converting between degrees and radians, and analyzing all sector properties essential for students, engineers, architects, designers, mathematicians, and anyone requiring accurate sector calculations for mathematics education, geometry problems, engineering design, architectural planning, circular construction, manufacturing, or problem-solving in education, design, and technical applications.
🥧 Sector Area Calculator
Calculate all sector properties
Calculate Sector Area
From radius and central angle
Calculate Arc Length
Length of the curved edge
Find Central Angle
From sector area and radius
Complete Sector Analysis
All properties at once
Understanding Circular Sectors
A circular sector is a pie-slice shaped region of a circle bounded by two radii and an arc. It consists of two straight edges (radii) and one curved edge (arc). The central angle is the angle between the two radii measured from the circle's center. Common examples include pizza slices, pie charts, and pendulum swings. A full circle is a 360° sector, a semicircle is a 180° sector, and a quarter circle is a 90° sector.
Sector Area Formulas
Basic Sector Area Formulas
Sector Area (angle in radians):
\[ A = \frac{1}{2}r^2\theta \]
Sector Area (angle in degrees):
\[ A = \frac{\theta}{360} \times \pi r^2 \]
Where:
\( A \) = sector area
\( r \) = radius
\( \theta \) = central angle
\( \pi \) ≈ 3.14159
Arc Length Formulas
Arc Length (angle in radians):
\[ L = r\theta \]
Arc Length (angle in degrees):
\[ L = \frac{\theta}{360} \times 2\pi r \]
Sector Perimeter:
\[ P = L + 2r = r\theta + 2r = r(\theta + 2) \]
Reverse Formulas
Angle from Sector Area (radians):
\[ \theta = \frac{2A}{r^2} \]
Angle from Arc Length (radians):
\[ \theta = \frac{L}{r} \]
Radius from Area and Angle:
\[ r = \sqrt{\frac{2A}{\theta}} \]
Angle Conversion
Degrees to Radians:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
Radians to Degrees:
\[ \text{degrees} = \text{radians} \times \frac{180}{\pi} \]
Step-by-Step Examples
Example 1: Calculate Sector Area from Radius and Angle
Problem: Find the sector area with radius 10 cm and central angle 60°.
Step 1: Identify the formula
\[ A = \frac{\theta}{360} \times \pi r^2 \]
Step 2: Substitute values
\[ A = \frac{60}{360} \times \pi \times 10^2 \]
Step 3: Simplify
\[ A = \frac{1}{6} \times \pi \times 100 = \frac{100\pi}{6} \]
Step 4: Calculate
\[ A \approx 52.36 \text{ cm}^2 \]
Answer: The sector area is approximately 52.36 cm².
Example 2: Calculate Arc Length
Problem: Find the arc length for radius 10 cm and central angle 90°.
Step 1: Use the arc length formula
\[ L = \frac{\theta}{360} \times 2\pi r \]
Step 2: Substitute and calculate
\[ L = \frac{90}{360} \times 2\pi \times 10 = \frac{1}{4} \times 20\pi \approx 15.71 \text{ cm} \]
Answer: The arc length is approximately 15.71 cm.
Example 3: Find Central Angle from Area
Problem: A sector has area 78.54 cm² and radius 10 cm. Find the central angle.
Method 1: Using degrees formula
\[ 78.54 = \frac{\theta}{360} \times \pi \times 100 \]
\[ \theta = \frac{78.54 \times 360}{100\pi} \approx 90° \]
Method 2: Using radians formula
\[ \theta = \frac{2A}{r^2} = \frac{2 \times 78.54}{100} \approx 1.571 \text{ rad} \approx 90° \]
Sector Measurements Reference Table
| Radius | Angle | Sector Area | Arc Length | Perimeter |
|---|---|---|---|---|
| 10 | 30° (π/6 rad) | 26.18 | 5.24 | 25.24 |
| 10 | 60° (π/3 rad) | 52.36 | 10.47 | 30.47 |
| 10 | 90° (π/2 rad) | 78.54 | 15.71 | 35.71 |
| 10 | 120° (2π/3 rad) | 104.72 | 20.94 | 40.94 |
| 10 | 180° (π rad) | 157.08 | 31.42 | 51.42 |
Common Sector Angles and Their Fractions
| Angle (Degrees) | Angle (Radians) | Fraction of Circle | Common Name |
|---|---|---|---|
| 30° | π/6 ≈ 0.524 | 1/12 | One-twelfth |
| 45° | π/4 ≈ 0.785 | 1/8 | One-eighth |
| 60° | π/3 ≈ 1.047 | 1/6 | One-sixth |
| 90° | π/2 ≈ 1.571 | 1/4 | Quarter circle |
| 120° | 2π/3 ≈ 2.094 | 1/3 | One-third |
| 180° | π ≈ 3.142 | 1/2 | Semicircle |
| 270° | 3π/2 ≈ 4.712 | 3/4 | Three-quarters |
| 360° | 2π ≈ 6.283 | 1 | Full circle |
Real-World Applications
Engineering & Design
- Gear design: Calculate tooth spacing and gear sector areas
- Cam mechanisms: Design circular cam profiles
- Pendulum analysis: Calculate pendulum swing areas
- Circular motion: Analyze rotational paths and coverage
Architecture & Construction
- Roof design: Calculate circular roof section areas
- Window arches: Determine curved window dimensions
- Circular stairs: Design spiral staircase sections
- Dome construction: Calculate dome segment areas
Data Visualization
- Pie charts: Calculate sector angles for data percentages
- Infographics: Design circular percentage displays
- Gauges and dials: Create circular measurement indicators
- Progress wheels: Display completion percentages
Food & Culinary
- Pizza slicing: Calculate slice sizes and servings
- Cake cutting: Determine equal portion sizes
- Pie serving: Calculate serving area per person
- Recipe scaling: Adjust circular food portions
Tips for Sector Calculations
Best Practices:
- Check angle units: Always verify degrees vs radians before calculating
- Convert correctly: Multiply by π/180 for degrees to radians
- Use fraction method: For degrees, think of (θ/360) as fraction of circle
- Remember π: Don't forget π in area formulas
- Sector vs circle: Sector is fraction of full circle area
- Arc vs chord: Arc is curved edge, chord is straight line
- Include units: Area in square units, arc length in linear units
Common Mistakes to Avoid
⚠️ Calculation Errors
- Wrong angle unit: Using degrees formula with radians or vice versa
- Forgetting ½ factor: Radian formula is ½r²θ, not r²θ
- Missing π: Degree formula needs πr², not just r²
- Wrong fraction: Use θ/360 for full calculation, not just θ
- Confusing arc and chord: Arc is curved distance, not straight
- Angle > 360°: Sector angle should be between 0° and 360°
- Negative angles: Use absolute value of angle
- Calculator mode: Ensure calculator set to correct angle mode
Frequently Asked Questions
How do you calculate the area of a sector?
Use formula A = (θ/360) × πr² for degrees or A = ½r²θ for radians. Think of sector as fraction of circle. Example: 60° angle, radius 10 cm. Sector is 60/360 = 1/6 of circle. Circle area = π(10)² = 314.16 cm². Sector area = 314.16/6 ≈ 52.36 cm². Or directly: A = (60/360) × π × 100 = 52.36 cm². For radians: convert 60° = π/3 rad, then A = ½ × 100 × π/3 = 52.36 cm². Both methods give same result.
What is the difference between arc length and sector area?
Arc length is the curved edge distance (one-dimensional), sector area is the surface inside (two-dimensional). Arc length = (θ/360) × 2πr in linear units (cm, m). Sector area = (θ/360) × πr² in square units (cm², m²). Example: radius 10 cm, angle 90°. Arc length = (90/360) × 2π × 10 = 15.71 cm. Sector area = (90/360) × π × 100 = 78.54 cm². Arc is perimeter, area is space. Both depend on same angle and radius.
How do you convert degrees to radians?
Multiply degrees by π/180. Formula: radians = degrees × π/180. Example: 60° = 60 × π/180 = π/3 ≈ 1.047 rad. Common conversions: 30° = π/6, 45° = π/4, 90° = π/2, 180° = π, 360° = 2π. Reverse: multiply radians by 180/π for degrees. Example: π/2 rad = (π/2) × 180/π = 90°. Radians are unitless—ratio of arc length to radius. One full circle = 2π radians = 360 degrees.
What is the perimeter of a sector?
Perimeter = arc length + two radii. Formula: P = L + 2r where L is arc length. Example: radius 10 cm, angle 60°. Arc length = (60/360) × 2π × 10 = 10.47 cm. Perimeter = 10.47 + 2(10) = 10.47 + 20 = 30.47 cm. Using radians: P = rθ + 2r = r(θ + 2). For 60° = π/3 rad: P = 10(π/3 + 2) = 10(1.047 + 2) = 30.47 cm. Not same as arc length—includes straight edges.
How do you find the central angle from sector area?
Rearrange area formula. For degrees: θ = (360 × A)/(πr²). For radians: θ = 2A/r². Example: area 52.36 cm², radius 10 cm. Using degrees: θ = (360 × 52.36)/(π × 100) = 18,849.6/314.16 ≈ 60°. Using radians: θ = (2 × 52.36)/100 = 104.72/100 = 1.047 rad = 60°. Essential for reverse engineering or when angle unknown but area measured. Verify result: recalculate area with found angle to check.
Can sector area be larger than circle area?
No! Sector is part of circle, so sector area ≤ circle area. Maximum sector is full circle (360° or 2π radians). If calculation gives sector area > circle area, error occurred—check angle units, formula, or arithmetic. Common mistake: using degrees in radian formula or vice versa. Example: radius 10 cm, circle area = 314.16 cm². Any sector must be ≤ 314.16 cm². For 60° sector: area = 52.36 cm² (about 1/6 of circle). Always verify sector area makes sense relative to full circle.
Key Takeaways
Understanding sector area calculations is essential for geometry, engineering, design, and data visualization. The formulas A = (θ/360) × πr² for degrees and A = ½r²θ for radians provide the foundation for all sector-related calculations, from basic area to complex design applications.
Essential principles to remember:
- Sector area (degrees): A = (θ/360) × πr²
- Sector area (radians): A = ½r²θ
- Arc length (degrees): L = (θ/360) × 2πr
- Arc length (radians): L = rθ
- Sector perimeter: P = L + 2r
- Degrees to radians: multiply by π/180
- Radians to degrees: multiply by 180/π
- Sector is fraction of circle
- Always verify angle units before calculating
- Area in square units, arc length in linear units
Getting Started: Use the interactive calculator at the top of this page to calculate sector area, arc length, central angle, and perimeter. Choose your calculation type, enter values in degrees or radians, and receive instant results with step-by-step solutions and detailed formulas. Perfect for students, engineers, designers, and anyone needing accurate sector calculations for education, engineering, or design projects.