Sector Area Calculator

Calculate the area of a sector using its radius and central angle.

Calculator

Enter Sector Dimensions

Guide

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

  1. Identify the given values: Determine what information you have (radius, angle, arc length, etc.).
  2. Convert units if necessary: Ensure angles are in the appropriate unit (degrees or radians).
  3. Select the appropriate formula: Choose the formula that matches your available information.
  4. Substitute values: Place your known values into the formula.
  5. Calculate the result: Perform the mathematical operations to find the area.
  6. 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.
Concept

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.

Formula

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)

Steps

How to Calculate Sector Area

  1. 1
    Measure the radius of the circle
  2. 2
    Measure the central angle in degrees
  3. 3
    Divide the central angle by 360
  4. 4
    Multiply by π (pi)
  5. 5
    Multiply by the square of the radius
Examples

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