Segment Area Calculator
Calculate the area of circular segments, chord length, and arc length with step-by-step solutions
Calculate Circular Segment Area
Distance from center to circle edge
Angle at circle center between two radii
Calculation Results
Example Calculations
Example 1: Using Radius and Angle
Given: Radius = 5 cm, Central angle = 60°
Convert angle: 60° = π/3 ≈ 1.047 radians
Formula: A = 0.5 × r² × (α - sin(α))
Calculation: A = 0.5 × 5² × (1.047 - sin(1.047)) = 0.5 × 25 × (1.047 - 0.866) = 2.27 cm²
Example 2: Using Radius and Height
Given: Radius = 10 m, Height = 3 m
Formula: A = r² × arccos((r-h)/r) - (r-h) × √(2×r×h - h²)
Calculation:
• arccos((10-3)/10) = arccos(0.7) ≈ 0.795 rad
• √(2×10×3 - 3²) = √51 ≈ 7.14
• A = 10² × 0.795 - 7 × 7.14 = 79.5 - 50.0 = 29.5 m²
Example 3: Partially Filled Pipe
Application: Water pipe with radius 20 cm, filled to height 8 cm
Given: r = 20 cm, h = 8 cm
Result: Cross-sectional water area ≈ 126.9 cm²
Use case: Calculate flow capacity and water volume
Segment vs Sector
Segment
Area bounded by chord and arc
= Sector - Triangle
Sector
Area bounded by two radii and arc
= 0.5 × r² × α
Chord
Straight line connecting two points on circle
Length = 2r × sin(α/2)
Real-World Applications
Architecture and construction design
Water volume in partially filled pipes
Road engineering and tunnel design
Mechanical engineering components
Sports field design and layouts
Understanding Circular Segments
What is a Circular Segment?
A circular segment is the region bounded by a chord and the arc of a circle. When you make a straight cut across a circle, you create two segments - a minor segment (smaller) and a major segment (larger).
Key Components
- •Chord: The straight line cutting the circle
- •Arc: The curved boundary of the segment
- •Height (Sagitta): Distance from chord midpoint to arc
- •Central Angle: Angle at center subtended by the arc
Formula Derivations
Method 1: Radius + Angle
Asegment = Asector - Atriangle
Asector = 0.5 × r² × α
Atriangle = 0.5 × r² × sin(α)
Result: A = 0.5 × r² × (α - sin(α))
Method 2: Radius + Height
First find the central angle:
α = 2 × arccos((r-h)/r)
Then apply integration method:
Result: A = r² × arccos((r-h)/r) - (r-h) × √(2rh - h²)
Practical Applications
🏗️ Engineering
Calculate cross-sectional areas in pipes, tanks, and structural components for flow analysis and load calculations.
🏛️ Architecture
Design arched windows, doorways, and decorative elements with precise area calculations for materials.
🌊 Hydraulics
Determine water volume in partially filled circular pipes and channels for flow rate calculations.