Circle & Sector Calculator

About the Circle & Sector Calculator

This calculator provides comprehensive circle geometry calculations, including circumference, area, sector arc length, sector area, chord length, and circular segment area. Every calculation includes detailed step-by-step derivations rendered with professional KaTeX math typesetting — great for students from elementary school through high school.

How to Use This Calculator

1. Choose a calculation mode: Circle, Sector, Chord & Segment, or Angle Conversion
2. Enter known values: radius (required) and central angle (mode dependent)
3. Choose the angle unit: degrees (°) or radians (rad)
4. Results update automatically, including circumference, area, arc length, and more
5. Tap "Show Steps" to see the full formula derivation and calculation process

Circle & Sector Formula Guide

Circle circumference C = 2πr, area A = πr², where r is the radius and π ≈ 3.14159

Sector arc length l = rθ (θ in radians), sector area A = (1/2)r²θ, or using degrees: A = (θ/360°)×πr²

Chord length c = 2r·sin(θ/2), segment area A = (1/2)r²(θ - sinθ), where θ is in radians

Angle conversion: 180° = π rad, so 1° = π/180 rad, and 1 rad = 180°/π ≈ 57.2958°

Frequently Asked Questions

What is the value of π (pi)?

π (Pi) is an irrational number approximately equal to 3.14159265358979... This calculator uses JavaScript's Math.PI constant, accurate to 15 decimal places.

How do I convert between degrees and radians?

The conversion relationship is 180° = π rad. Degree to radian: θ(rad) = θ(°) × π/180°. Radian to degree: θ(°) = θ(rad) × 180°/π. For example, 90° = π/2 rad and 1 rad ≈ 57.2958°.

What is a circular segment?

A circular segment is the region between a chord and the arc it cuts off. Segment area = sector area − triangle area. Formula: A = (1/2)r²(θ - sinθ), where θ is the central angle in radians.

Why does the arc length formula require radians?

The arc length formula l = rθ only works when θ is in radians, by the definition of a radian — the angle subtended when the arc length equals the radius. When using degrees, convert to radians first, or use l = (θ/360°)×2πr.

How accurate are the results?

This calculator uses JavaScript's Math.PI (accurate to 15 decimal places) and trigonometric functions. Circle, circumference, area, and sector calculations have error under 0.001%, while chord and segment calculations have error under 0.1%. All formulas are verified against Khan Academy and Wolfram MathWorld.

Real-World Applications

• Architecture: area and perimeter of circular buildings, arches, and domes
• Manufacturing: circular parts, sector gears, and curved material cutting calculations
• Math education: teaching circle area, sector area, and radian measure at every grade level
• Everyday life: pizza slice area, circular garden design, cake slice portions
• Measurement: arc-length surveying, circular track distance, area estimation

Which Education Levels Is This For?

• Elementary (grades 5-6): circle circumference and area formulas (C = 2πr, A = πr²)
• Middle school (grades 7-9): sector area, arc length, central angle concepts
• High school: radian measure, chord length, segment area, trigonometric applications
• College/engineering: precise calculation, formula derivation, real-world engineering use