Sector Area Calculator

Calculate all properties of a circle sector including area, arc length, chord length, and central angle. Works with radius and angle, radius and arc length, or other combinations.

Input Parameters

r α L

Results

Enter values and click Calculate to see results.

What is a Sector of a Circle?

A sector is a geometric figure bounded by two radii and the included arc of a circle. Think of it like a slice of pie or pizza – that's a sector! Sectors are commonly visualized in pie charts, where a circle is divided into several sectors to show the proportion of each segment.

A sector is defined by:

  • Radius (r) – The distance from the center to the edge of the circle
  • Central Angle (α) – The angle formed by the two radii at the center
  • Arc Length (L) – The curved distance along the circle's edge
  • Chord Length (c) – The straight-line distance between the two endpoints

Sector Area Formula

The formula for calculating the area of a sector is elegantly simple:

A = r² × α / 2

(where α is in radians)

If you prefer working with degrees, use this alternative formula:

A = r² × θ × π / 360

(where θ is in degrees)

Where Does This Formula Come From?

The sector area formula is derived using proportions. Here's the logic:

  1. The area of a full circle is A = πr²
  2. A full circle has an angle of 2π radians (or 360°)
  3. A sector is just a portion of the circle based on its central angle
  4. Using proportions: If 2π → πr², then α → ?
  5. Solving: Sector Area = α × πr² / 2π = α × r² / 2

Special Cases

Semicircle (Half Circle)

A semicircle is a sector with a central angle of 180° (π radians):

Semicircle Area = πr² / 2

Quadrant (Quarter Circle)

A quadrant is a sector with a central angle of 90° (π/2 radians):

Quadrant Area = πr² / 4

Related Formulas

Arc Length

The arc length of a sector can be calculated using:

L = r × α (α in radians)

L = r × θ × π / 180 (θ in degrees)

Chord Length

The chord length (straight line between endpoints) is:

c = 2r × sin(α/2)

Finding Central Angle from Area

If you know the sector area and radius, you can find the central angle:

α = 2A / r² (result in radians)

θ = 360A / (πr²) (result in degrees)

Real-Life Applications

Sector area calculations are more common in everyday life than you might think:

  • 🍰 Cake Portions – Planning a birthday party? Calculate the size of each cake slice to ensure everyone gets a fair share and nobody goes hungry!
  • 🍕 Pizza Slices – Ever noticed that pizza slices are sectors? Calculate which pizza size gives you the best value per slice, or determine how much crust you'll get with each piece.
  • 👗 Sewing Circle Skirts – Creating a circle skirt often involves cutting a sector from fabric. Calculate the right dimensions for your pattern.
  • 📊 Data Visualization – Pie charts use sectors to represent data proportions. Understanding sector areas helps you interpret these visual representations accurately.
  • 🏗️ Architecture and Design – Circular windows, arches, and decorative elements often use sector shapes in their design.
  • 🎯 Sports and Games – Dartboards, archery targets, and many game boards use sectors to divide playing areas.
  • 📐 Advanced Geometry – Sector calculations are fundamental in finding the surface area of cones and other 3D shapes.

Step-by-Step Examples

Example 1: Finding Sector Area from Radius and Angle

Given: Radius = 10 cm, Central angle = 60°

Solution:

  1. Convert angle to radians: α = 60° × π/180° = π/3 ≈ 1.047 radians
  2. Apply formula: A = r² × α / 2
  3. Calculate: A = 10² × 1.047 / 2 = 52.36 cm²

Alternative (using degrees): A = 10² × 60 × π / 360 = 52.36 cm²

Example 2: Finding Central Angle from Area and Radius

Given: Radius = 8 inches, Sector Area = 50.27 in²

Solution:

  1. Use formula: α = 2A / r²
  2. Calculate: α = 2 × 50.27 / 8² = 1.571 radians
  3. Convert to degrees: θ = 1.571 × 180 / π ≈ 90°

Result: This is a quadrant (quarter circle)!

Example 3: Finding Radius from Area and Angle

Given: Sector Area = 78.54 cm², Central angle = 120°

Solution:

  1. Convert angle: α = 120° × π/180° = 2.094 radians
  2. Rearrange formula: r² = 2A / α
  3. Calculate: r² = 2 × 78.54 / 2.094 = 75
  4. Take square root: r = √75 ≈ 8.66 cm

Frequently Asked Questions

What's the difference between a sector and a segment?

A sector is bounded by two radii and an arc, like a pizza slice. A segment is the area between a chord and the arc – it's what's left when you cut off a sector with a straight line that doesn't go through the center.

Can a sector be larger than a semicircle?

Yes! A sector can have a central angle anywhere from 0° to 360°. When the angle is greater than 180° (but less than 360°), you have a reflex sector or major sector. The smaller sector (less than 180°) is called a minor sector.

What is the area of a 90° sector with radius 1?

This is a quadrant. Using the formula: A = r² × θ × π / 360 = 1² × 90 × π / 360 = π/4 ≈ 0.785 square units. This is exactly one-quarter of the full circle area (π).

How do I convert between radians and degrees?

Use these conversion formulas:

  • Radians to Degrees: degrees = radians × 180 / π
  • Degrees to Radians: radians = degrees × π / 180

Remember: π radians = 180°, so 2π radians = 360° (full circle).

What if I only know the arc length and radius?

No problem! First find the central angle: α = L / r (in radians), then use the sector area formula: A = r² × α / 2. This calculator handles this scenario automatically!

Tips for Using This Calculator

  • Choose the calculation mode based on what information you have available
  • Make sure to use consistent units for all measurements
  • The calculator accepts both degrees and radians – choose what's most convenient for you
  • Use the visualization to verify your results make sense
  • For very large or small values, consider using scientific notation

💡 Pro Tip

When working with sectors, always double-check whether you're using radians or degrees. This is the most common source of calculation errors! Most calculators and programming languages use radians by default for trigonometric functions.

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