Segment of Circle Calculator

Calculate area, arc length, and height of circular segments using radius, angle, or chord length

Input Parameters

Enter your parameters and click "Calculate" to see results.
The diagram shows the circle with the calculated segment highlighted.
Calculation steps will appear here after you click "Calculate".

About Circular Segments

What is a Circular Segment?

A circular segment is a region of a circle bounded by a chord and the arc subtended by that chord. There are two types:

  • Minor Segment: The smaller area when the central angle is less than 180°
  • Major Segment: The larger area when the central angle is less than 180°
Practical Applications
  • Architectural design of arches and windows
  • Engineering calculations for curved structures
  • Land surveying for circular plots
  • Manufacturing of circular parts with cutouts
Example Calculations
Example 1: Angle-based Calculation

For a circle with radius 5m and central angle 60°:

  • Segment Area: 1.87 m²
  • Arc Length: 5.24 m
  • Segment Height: 0.67 m
Example 2: Chord-based Calculation

For a circle with radius 8m and chord length 10m:

  • Segment Area: 14.18 m²
  • Central Angle: 77.36°
  • Segment Height: 1.79 m

Complete Guide to Circular Segments

1. Understanding Circular Segments

A circular segment is a fundamental geometric shape formed by cutting a circle with a straight line (chord). The segment is defined by three key elements:

  • Chord: The straight line connecting two points on the circle
  • Arc: The curved portion of the circle's circumference between the chord endpoints
  • Sagitta: The height of the segment, measured perpendicular from chord to arc

When the central angle equals 180°, the segment becomes a semicircle. At 360°, it's the full circle.

2. Formula Breakdown and Derivation

Primary Formula: Segment Area (Angle Method)

A = (r²/2) × (θ - sinθ)

  • r: Circle radius (constant distance from center to circumference)
  • θ: Central angle in radians (1 radian ≈ 57.2958 degrees)
  • sinθ: Trigonometric sine function of the angle

Derivation: This formula comes from subtracting the area of the triangular portion (formed by radii and chord) from the sector area.

Key Relationships

  • Chord Length: c = 2r × sin(θ/2) - Derived from the Law of Sines in the isosceles triangle
  • Sagitta (Segment Height): h = r × (1 - cos(θ/2)) - Represents the radius minus the apothem distance
  • Arc Length: L = r × θ - Direct proportion between angle and circumference

3. Step-by-Step Calculation Example

Problem: Calculate all parameters for a circular segment with radius 10m and central angle 60°
Step 1: Convert angle to radians

θ = 60° × (π/180) = 1.0472 radians

Step 2: Calculate chord length

c = 2 × 10 × sin(1.0472/2) = 20 × sin(0.5236) = 20 × 0.5 = 10.0000 m

Step 3: Calculate sagitta (segment height)

h = 10 × (1 - cos(0.5236)) = 10 × (1 - 0.8660) = 1.3397 m

Step 4: Calculate arc length

L = 10 × 1.0472 = 10.4720 m

Step 5: Calculate segment area

A = (10²/2) × (1.0472 - sin(1.0472)) = 50 × (1.0472 - 0.8660) = 9.0586 m²

Verification: For small angles (θ < 90°), the segment area approximates (c × h)/3, which gives (10 × 1.3397)/3 = 9.0647 m² (close match).

4. Professional Applications and Real-World Usage

Architecture & Construction
  • Arch Design: Calculating materials for arched doorways, windows, and bridges
  • Dome Construction: Determining panel sizes for spherical structures
  • Tunnel Engineering: Cross-sectional area calculations for circular tunnels
  • Staircase Design: Curved handrails and spiral staircase planning
Engineering & Manufacturing
  • Mechanical Parts: Gears, cams, and circular cutouts in machinery
  • Pipe Systems: Flow area in partially filled pipes
  • Tank Design: Liquid volume in horizontal cylindrical tanks
  • Aerospace: Wing cross-sections and fuselage segments
Surveying & Geography
  • Land Parcels: Area calculations for circular plot segments
  • Irrigation: Water coverage area from circular sprinklers
  • Satellite Coverage: Earth surface area covered by satellite beams
  • Cartography: Map projections and spherical geometry
Design & Art
  • Graphic Design: Creating circular logos and interface elements
  • Woodworking: Calculating material for curved furniture pieces
  • Stained Glass: Pattern cutting for circular window sections
  • Jewelry Making: Precious metal calculations for circular designs

5. Measurement Guidance and Best Practices

Input Accuracy Tips
  • Radius Measurement: Measure from the exact center to the circumference. Use calipers for precision.
  • Angle Measurement: For field measurements, use a protractor or digital angle finder.
  • Chord Measurement: Straight line distance between two points. Ensure tape is taut and level.
  • Unit Consistency: Always use the same unit system throughout calculations.
Common Measurement Scenarios:
Scenario Recommended Method Precision Tips
Complete circle available Measure diameter, divide by 2 for radius Measure at multiple points, take average
Only segment visible Measure chord and sagitta, calculate radius Use formula: r = (c²/8h) + (h/2)
Field surveying Use chord length and central angle Use theodolite for angle measurement
Manufacturing CAD measurements from drawings Account for material thickness and tolerances

6. Units and Conversion Reference

Linear Units
  • Meter (m): SI standard unit, recommended for engineering
  • Centimeter (cm): 1 m = 100 cm, useful for smaller segments
  • Millimeter (mm): 1 m = 1000 mm, precision manufacturing
  • Inch (in): 1 in = 2.54 cm, imperial system
  • Foot (ft): 1 ft = 0.3048 m, construction industry
Area Units
  • Square meter (m²): Standard SI area unit
  • Square centimeter (cm²): 1 m² = 10,000 cm²
  • Square millimeter (mm²): 1 m² = 1,000,000 mm²
  • Square foot (ft²): 1 m² ≈ 10.7639 ft²
  • Square inch (in²): 1 m² ≈ 1550 in²
Unit Conversion Tip

When converting area units, remember that area conversions involve squaring the linear conversion factor. For example: 1 m = 100 cm, so 1 m² = (100)² cm² = 10,000 cm².

7. Common Mistakes and How to Avoid Them

Common Error Why It Happens How to Prevent
Using degrees instead of radians Forgetting trigonometric functions require radians Always convert: radians = degrees × π/180
Confusing segment with sector Similar shapes but different formulas Segment has chord, sector has radii to arc ends
Incorrect major/minor selection Not understanding angle relationships Major segment = circle area - minor segment area
Unit inconsistency Mixing different measurement systems Convert all inputs to same unit before calculating
Chord > diameter Physically impossible in a circle Validate: chord length ≤ 2 × radius
Precision loss Rounding intermediate values Carry full precision through calculations

8. Precision and Rounding Guidelines

Significant Figures Rules
  • Input Precision: Results should not be more precise than your least precise measurement
  • Engineering Standard: Typically 3-4 significant figures for most applications
  • Construction: Round to nearest 1/8 inch or 1 mm for practical use
  • Manufacturing: May require precision to 0.001 mm or better
When to Round:
  1. Never round intermediate values - carry full precision through calculations
  2. Round only the final result to appropriate significant figures
  3. Consider the application - architectural vs. mechanical precision differs
  4. Include error margins for critical applications (e.g., +2%)
Example: Rounding for Different Applications

Calculated area: 9.058647 m²

  • Land survey: 9.06 m² (2 decimal places)
  • Construction estimate: 9.1 m² (1 decimal place)
  • Precision machining: 9.0586 m² (4 decimal places)
  • Academic reporting: 9.05865 m² (match input precision)

9. Educational Notes for Students

Learning Objectives
  • Understand the relationship between chord, arc, and segment
  • Apply trigonometric functions to geometric problems
  • Convert between degrees and radians correctly
  • Derive segment formulas from first principles
  • Solve real-world problems using segment geometry
Study Tips
  • Draw diagrams for every problem
  • Memorize the unit circle values (0°, 30°, 45°, 60°, 90°)
  • Practice deriving formulas instead of just memorizing
  • Use dimensional analysis to check your work
  • Work through problems both numerically and symbolically
Common Exam Questions
  1. Given radius and chord, find the area of the segment
  2. Given segment area and radius, find the central angle
  3. Prove that segment area = (r²/2)(θ - sinθ)
  4. Find the ratio of segment area to circle area for given angle
  5. Application problems involving arches, tanks, or land areas

10. Accessibility and Usability Notes

Accessibility Features
  • Keyboard Navigation: All calculator functions accessible via keyboard
  • Screen Reader Compatible: Proper ARIA labels and semantic HTML
  • High Contrast Mode: Dark mode option for low vision users
  • Zoom Friendly: Responsive design maintains functionality at 200% zoom
  • Color Blind Friendly: Color choices avoid red-green confusion
Device Compatibility
Desktop/Laptop

Full functionality with mouse and keyboard

Tablet

Touch-optimized inputs and responsive layout

Smartphone

Mobile-first design with simplified interface

Offline Usage

This calculator works entirely in your browser - no server communication needed after initial load. You can use it without an internet connection once the page is loaded.

11. Related Geometry Tools

Circle Sector Calculator

Calculate area and perimeter of circle sectors

Arc Length Calculator

Find arc length from radius and angle

Chord Length Calculator

Calculate chord length from radius and angle

Circle Area Calculator

Calculate area, circumference, diameter

Annulus Calculator

Calculate area between two concentric circles

Ellipse Segment Calculator

Advanced calculations for elliptical segments

12. Professional Disclaimer and Notes

Important Disclaimers
  • Educational Purpose: This tool is designed for educational and planning purposes only
  • Professional Verification: Critical applications require verification by licensed professionals
  • Measurement Accuracy: Results depend on input accuracy - "garbage in, garbage out"
  • Rounding Differences: Different calculation methods may yield slightly different results
  • No Liability: We assume no responsibility for decisions made based on these calculations
Recommended Professional Standards
  • Construction: ASME Y14.5 - Dimensioning and Tolerancing
  • Surveying: FGDC Geospatial Positioning Accuracy Standards
  • Manufacturing: ISO 2768 - General Tolerances
  • Education: NCTM Principles and Standards for School Mathematics
Version Information

Current Version: 2.1.0

Last Updated: October 2025

Algorithm Source: Standard geometric formulas with double-precision floating point

Browser Compatibility: Chrome 90+, Firefox 88+, Safari 14+, Edge 90+

Calculation Method: Pure JavaScript client-side computation

How to Cite This Tool

APA Style: ToolsRail Geometry. (2026). Segment of Circle Calculator. Retrieved from https://www.toolsrail.com/geometry/segment-of-circle-calculator.php

MLA Style: "Segment of Circle Calculator." ToolsRail Geometry, ToolsRail, 2026, www.toolsrail.com/geometry/segment-of-circle-calculator.php.