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Circle Chord Calculator
Calculate chord length, central angle, and segment properties
📚 Examples, Rules & Help
⚡Quick Examples of Circle Chord
🔍How to Calculate Circle Chord
Convert Angle to Radians
If angle is in degrees: θ(radians) = θ(degrees) × π/180. If already in radians, use directly.
Calculate Chord Length
Chord = 2r × sin(θ/2). This uses the properties of the isosceles triangle formed by two radii and the chord.
Calculate Segment Area
Segment Area = Sector Area - Triangle Area. The segment is the 'cap' between the chord and the arc.
Find Sagitta and Distance
Sagitta = r(1 - cos(θ/2)) is the height of the segment. Distance to chord = r × cos(θ/2).
🌍Real-World Applications
🏗️ Architecture
Arch design and span calculations
🌉 Engineering
Bridge design, curved structures
🏭 Manufacturing
Circular cuts, curved edges
🔬 Optics
Lens design, curved mirrors
❓Frequently Asked Questions
What is a chord in a circle?
A chord is a straight line segment that connects two points on a circle. The diameter is the longest possible chord, passing through the center of the circle.
How do you calculate chord length?
Chord length = 2r × sin(θ/2), where r is the radius and θ is the central angle in radians. You can also use the chord length to find the central angle.
What's the difference between a chord and an arc?
A chord is the straight-line distance between two points on a circle, while an arc is the curved distance along the circle's circumference between the same two points.
What is the relationship between chord length and central angle?
As the central angle increases, the chord length increases, but not linearly. The relationship follows the sine function: chord = 2r sin(θ/2).
How do you find the central angle from chord length?
Central angle θ = 2 × arcsin(chord length / (2r)). This gives the angle in radians; multiply by 180/π for degrees.
What is a chord segment?
A chord segment (or circular segment) is the area between a chord and the arc it subtends. It's calculated as sector area minus triangle area.
🎯Common Use Cases
Construction & Architecture
- Arch span calculations for bridges and doorways
- Window and door frame design for curved openings
- Roof truss calculations for curved ceiling designs
- Foundation layout for circular or curved structures
Manufacturing & Engineering
- Gear tooth spacing and mechanical part design
- Pipe bending calculations for plumbing and HVAC
- Sheet metal forming for curved automotive parts
- Lens grinding and optical component manufacturing
Special Mathematical Cases
- Diameter: θ = 180°, chord = 2r (maximum possible)
- Right angle: θ = 90°, chord = r√2 ≈ 1.414r
- Equilateral triangle: θ = 60°, chord = r
- Small angles: chord ≈ arc length (useful approximation)
💡Calculator Tips & Best Practices
💡Understanding Chord vs Arc
A chord is always shorter than its corresponding arc. The difference becomes more pronounced as the central angle increases.
📏Maximum Chord Length
The diameter is the longest possible chord in a circle. Any chord with a 180° central angle equals the diameter (2r).
⭐Small Angle Approximation
For very small central angles (< 10°), the chord length closely approximates the arc length. This is useful for quick estimates.
⚠️Radian vs Degree Calculations
Always ensure your angle is in the correct unit. Most formulas expect radians; convert degrees using: radians = degrees × π/180.
📝Sagitta Applications
The sagitta (height of chord segment) is crucial in architecture and engineering for arch design and structural calculations.
⭐Precision for Construction
For construction projects, calculate chord lengths to at least 3 decimal places to ensure proper fit and structural integrity.
📚 References & Further Reading
Educational resource on chord properties and calculations
External Link
Precision measurement standards for geometric calculations
External Link
Note: These references provide additional mathematical context and verification of the formulas used in this calculator.