🥧 Sector Area Calculator
Calculate the area of a circular sector (pie slice)
How to Use This Calculator
Enter the Radius
Input the radius of the circle. This is the distance from the center to any point on the circle's edge.
Select Angle Unit
Choose whether your angle is in degrees or radians. Most commonly, angles are in degrees.
Enter the Central Angle
Input the central angle that defines the sector. For degrees, use values between 0 and 360.
Calculate
Click "Calculate Sector Area" to get the sector area and related measurements.
Formula
Sector Area = (1/2) × r² × θ
Where θ must be in radians
Alternative Formula (using degrees):
Sector Area = (θ/360°) × πr²
This formula uses degrees directly: multiply full circle area by the fraction of the circle
Where:
- r = radius of the circle
- θ = central angle in radians
- π ≈ 3.14159...
Example 1: Find sector area for r = 5 units, θ = 90°
Method 1 (radians): θ = 90° × (π/180) = π/2 ≈ 1.5708 rad
Area = (1/2) × 5² × 1.5708 = (1/2) × 25 × 1.5708 = 19.635 square units
Method 2 (degrees): Area = (90/360) × π × 5² = (1/4) × 78.54 = 19.635 square units
Example 2: Find sector area for r = 10 units, θ = 2.5 radians
Area = (1/2) × 10² × 2.5 = (1/2) × 100 × 2.5 = 125 square units
Example 3: Find sector area for r = 3 units, θ = 180° (semicircle)
Area = (180/360) × π × 3² = (1/2) × 28.27 = 14.137 square units
This is exactly half the circle, as expected for 180°.
About Sector Area Calculator
The Sector Area Calculator finds the area of a circular sector, which is the "pie slice" shaped region bounded by two radii and an arc. It's useful for calculating partial areas of circles.
When to Use This Calculator
- Geometry: Calculate sector areas in circle problems
- Engineering: Design circular components with partial coverage
- Architecture: Calculate material needed for curved segments
- Education: Learn and practice sector area concepts
- Design: Plan circular patterns and pie charts
- Real Estate: Calculate area of circular land parcels
Why Use Our Calculator?
- ✅ Flexible Units: Works with degrees or radians
- ✅ Comprehensive Output: Shows sector area, full circle area, fraction, and arc length
- ✅ Educational: Displays formulas and calculation steps
- ✅ Accurate: Precise mathematical calculations
- ✅ Instant Results: Calculate immediately
- ✅ 100% Free: No registration required
Understanding Sector Area
A sector is like a slice of pie. Its area is proportional to the central angle. A full circle (360°) has area πr², so a sector with angle θ has area proportional to θ/360°.
- Semicircle (180°): Area = πr²/2 (half the circle)
- Quarter circle (90°): Area = πr²/4 (one-fourth of the circle)
- Full circle (360°): Area = πr² (the entire circle)
- The sector area grows linearly with the angle
Real-World Applications
Pizza Slices: Calculate the area of individual pizza slices. If a 12-inch pizza (r=6) is cut into 8 slices, each 45° sector has area ≈ 14.14 in².
Land Surveying: Determine the area of circular land parcels or sections of circular properties.
Manufacturing: Calculate material needed for curved segments in circular designs.
Data Visualization: Create pie charts where each sector represents a percentage of the whole.
Frequently Asked Questions
What is the difference between a sector and a segment?
A sector includes the area from the center to the arc (pie slice shape). A segment is the area between a chord and the arc (doesn't include the center). Sector area = (1/2)r²θ, while segment area requires subtracting a triangle.
Why do I need to convert degrees to radians?
The formula Area = (1/2)r²θ works when θ is in radians. If you have degrees, convert using: radians = degrees × (π/180). Alternatively, use the formula Area = (θ/360°) × πr² which works directly with degrees.
How do I find sector area if I only know the arc length?
If you know arc length (L) and radius (r), first find the angle: θ = L/r (in radians). Then use Area = (1/2)r²θ. Or use Area = (1/2) × r × L directly.
Can the angle be greater than 360°?
Technically yes, but it represents more than one full rotation. For practical purposes, angles are usually between 0° and 360°. Our calculator accepts values up to 360°.
What's the relationship between sector area and arc length?
Sector Area = (1/2) × r × Arc Length. If you know the arc length, you can directly calculate area without finding the angle first. This formula is simpler and shows the direct relationship.
How do I calculate multiple sectors?
Calculate each sector separately and add them together. Make sure all angles are in the same unit (degrees or radians) and sum to 360° (or 2π radians) if they cover the full circle.