Sector Area Calculator
Calculate the area of a sector using its radius and central angle.
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Table of Contents
Comprehensive Guide to Sector Area
What is a Circle Sector?
A sector of a circle is a pie-shaped portion that consists of a region bounded by two radii and an arc. Imagine cutting a pizza into slices—each slice represents a sector. The sector is formed when two radii extend from the center and connect to an arc on the circle's circumference.
Types of Sectors
- Minor Sector: A sector with a central angle less than 180° (π radians).
- Major Sector: A sector with a central angle greater than 180° (π radians).
- Semicircular Sector: A sector with a central angle of exactly 180° (π radians).
- Quadrant: A sector with a central angle of exactly 90° (π/2 radians), representing one-quarter of a circle.
Key Components of a Sector
- Radius (r): The distance from the center of the circle to any point on its circumference.
- Central Angle (θ): The angle formed by the two radii at the center of the circle.
- Arc Length (s): The portion of the circumference enclosed by the sector.
Sector Area Formulas
When angle is in degrees:
Area = (θ/360°) × π × r²
When angle is in radians:
Area = (θ/2) × r²
Related Sector Measurements
Arc Length (s):
s = (θ/360°) × 2πr (degrees) or s = θ × r (radians)
Perimeter of a Sector:
Perimeter = 2r + s (Two radii + arc length)
Special Cases
Semicircle (θ = 180°):
Area = πr²/2
Quadrant (θ = 90°):
Area = πr²/4
Real-World Applications
- Architecture and Design: Calculating materials needed for circular features, fan-shaped windows, or arched doorways.
- Food Industry: Determining portion sizes for pizza, pie, or cake slices.
- Engineering: Designing gears, cam mechanisms, and circular components.
- Landscaping: Planning circular garden sections or irrigation systems with specific coverage angles.
- Navigation and Geography: Calculating radar coverage areas or satellite signal ranges.
Step-by-Step Problem Solving
- Identify the given values: Determine what information you have (radius, angle, arc length, etc.).
- Convert units if necessary: Ensure angles are in the appropriate unit (degrees or radians).
- Select the appropriate formula: Choose the formula that matches your available information.
- Substitute values: Place your known values into the formula.
- Calculate the result: Perform the mathematical operations to find the area.
- Verify your answer: Check if your result is reasonable based on the given values.
Common Mistakes to Avoid
- Forgetting to convert between degrees and radians when using formulas.
- Confusing the formulas for sector area and segment area.
- Overlooking the units of measurement in the final answer.
- Using diameter instead of radius in calculations.
What is Area?
The area of a sector is the amount of space enclosed within a portion of a circle, bounded by two radii and an arc. It's measured in square units and represents the two-dimensional space that the sector occupies. A sector is a pie-shaped portion of a circle.
Sector Area Formula
Formula
A = (θ/360) × π × r²
where θ is the central angle in degrees, r is the radius, and π is pi (approximately 3.14159)
How to Calculate Sector Area
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1Measure the radius of the circle
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2Measure the central angle in degrees
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3Divide the central angle by 360
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4Multiply by π (pi)
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5Multiply by the square of the radius
Practical Examples
Example
A sector has a radius of 5 units and a central angle of 60 degrees.
A = (θ/360) × π × r²
A = (60/360) × π × 5²
A = (1/6) × π × 25
A ≈ 13.09 square units