Sector Area Calculator
Calculate sector area, arc length, chord length, and other properties
Calculate Sector Properties
Angle between the two radii
Distance from center to circumference
Types of Sectors
Quadrant
Central angle = 90°
Area = πr²/4
Semicircle
Central angle = 180°
Area = πr²/2
Regular Sector
Any other angle
Area = (α × r²)/2
Key Formulas
Sector Area
A = (α × r²) / 2
α in radians
Arc Length
L = α × r
α in radians
Chord Length
c = 2r × sin(α/2)
α in radians
Angle Conversion
rad = deg × π/180
degrees to radians
Quick Examples
Pizza Slice
Pizza radius: 8 inches
Slice angle: 45°
Slice area: ≈ 25.1 in²
Pie Chart
Chart radius: 5 cm
Section: 25% (90°)
Section area: ≈ 19.6 cm²
Understanding Circle Sectors
What is a Sector?
A sector is a geometric figure bounded by two radii and the included arc of a circle. Think of it as a "slice" of a circle, like a piece of pie or pizza.
Key Components
- •Central Angle (α): Angle between the two radii
- •Radius (r): Distance from center to edge
- •Arc: Curved edge of the sector
- •Chord: Straight line connecting arc endpoints
Formula Derivation
The sector area formula comes from proportions. Since a full circle has area πr² and angle 2π radians, a sector with angle α has proportional area.
Step-by-step:
1. Full circle: 2π radians → πr² area
2. Sector: α radians → ? area
3. Proportion: α/2π = Sector Area/πr²
4. Solve: Sector Area = (α × πr²)/2π
5. Simplify: Sector Area = (α × r²)/2
Real-World Applications
- • Calculating cake or pizza slice sizes
- • Designing pie charts and data visualizations
- • Engineering circular components
- • Architecture with curved elements
- • Land surveying and area calculations