Sector Area Calculator
Calculate the area, radius, or central angle of a circular sector
Distance from center to edge
Angle at the center of the circle
Area of the pie slice portion
Sector Calculation Results
Sector Area
Radius
Angle
Arc Length
Length along the arc
Chord Length
Straight-line distance
Full Circle Area
Complete circle area
Angle (Radians)
Angle in radians
Formulas Used
Sector Area
A = ½r²θ
Arc Length
L = rθ
Chord Length
c = 2r·sin(θ/2)
Note: θ must be in radians for these formulas
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About Sector Area Calculator
What is Sector Area?
A sector is a "pie slice" shaped portion of a circle, bounded by two radii and the arc between them. The sector area is the amount of space enclosed within this pie slice. It's one of the fundamental measurements in circular geometry with applications in engineering, architecture, and data visualization.
How to Use This Calculator
- Select Mode: Choose between finding Sector Area, Radius, or Angle.
- Enter Values: Input the known measurements for your circular sector.
- Select Units: For angles, toggle between Degrees and Radians.
- Review Results: See the calculated values along with related arc and chord data.
Understanding Your Results
Primary Metric
Sector Area: The calculated area enclosed by the sector. This is the main result in finding sector area mode.
Secondary Metrics
Radius: The distance from the center to the circle's edge. Angle: The central angle in degrees or radians. Arc Length: The curved distance along the sector's outer edge. Chord Length: The straight-line distance between the arc endpoints.
Sector Area Formulas
Primary Formula
Sector Area = ½ × Radius² × Angle (in radians)
A = ½r²θ
Where:
- A = Sector area
- r = Radius of the circle
- θ = Central angle in radians
Alternative Formula (with degrees)
If your angle is in degrees:
A = (θ/360) × πr²
Derived Formulas
| Find | Formula |
|---|---|
| Sector Area | A = ½r²θ |
| Radius | r = √(2A/θ) |
| Angle | θ = 2A/r² |
Frequently Asked Questions (FAQ)
What is the difference between sector area and segment area?
Sector area includes the full pie slice from center to arc. Segment area is only the region between the chord and the arc (sector minus the triangle). This calculator computes sector area.
Why must the angle be in radians for the formula?
The formula A = ½r²θ only works when θ is in radians because radians provide a natural ratio-based measurement. This calculator automatically converts degrees to radians when needed.
How does sector area relate to the full circle?
The sector area is proportional to its central angle. A sector with a 90° angle has ¼ (25%) of the full circle's area. The formula effectively calculates: (angle/full rotation) × π × radius².
What's the sector area of a semicircle?
For a semicircle (180° or π radians), the sector area is exactly half the circle's area: A = ½πr²
Can sector area be larger than the circle area?
No, the maximum sector area (at 360° or 2π radians) equals the full circle area (πr²). Angles greater than a full rotation are not valid for standard sector calculations.
Practical Applications
- Pizza Slices: Calculate how much pizza each person gets
- Pie Charts: Determine area proportions in data visualization
- Engineering: Design gears, cams, and circular components
- Architecture: Calculate curved floor sections or dome portions
- Land Surveying: Measure curved boundary areas
Important Notes
- Always ensure your angle unit (degrees or radians) is correctly selected
- The radius and areas will use consistent square units
- For very small angles, sector area approaches a triangle's area
Disclaimer: This calculator is for informational purposes only and should not replace professional geometric or engineering calculations.