Chord Length Calculator
Calculate Chord Length
Calculate chord length of a circle from radius and central angle or from radius and perpendicular distance from center.
Chord Length Result
10.00 units
Circle Visualization
Formula Used:
Where: c = chord length, r = radius, θ = central angle in radians
Step-by-Step Calculation:
Sagitta (Chord Height)
Sagitta = r - √(r² - (c/2)²) = 10 - √(100 - 25)
Also called versine or chord height
Angle Conversions
Chord to Diameter Ratio: 0.5 (50%)
Arc Length: 10.47 units
Segment Area: 9.06 square units
Application: Architecture, engineering, circular segment calculations
For a circle with radius 10 units and central angle 60°, the chord length is 10 units. This chord divides the circle into two segments and is perpendicular to the radius at its midpoint.
What is a Chord?
A chord is a straight line segment whose endpoints both lie on a circle's circumference. The chord length is the distance between these two points along the straight line. Important properties include: the longest chord is the diameter, chords equidistant from the center are equal in length, and the perpendicular from the center bisects the chord.
Chord Formulas and Properties
From Radius & Angle
Most common formula
Requires central angle
From Radius & Distance
Using perpendicular distance
Pythagorean theorem
From Arc Length
Two-step calculation
Arc length known
Sagitta Formula
Chord height calculation
Also called versine
Special Chord Cases
Chord Length Formulas
1. From Radius and Central Angle
Where:
- c = chord length
- r = radius of the circle
- θ = central angle (in radians)
- For degrees: c = 2 × r × sin(θ° × π/360)
2. From Radius and Perpendicular Distance
Where:
- c = chord length
- r = radius of the circle
- d = perpendicular distance from center to chord
- Derived from Pythagorean theorem
3. From Chord and Sagitta
Where:
- r = radius of the circle
- c = chord length
- h = sagitta (height of circular segment)
Common Chord Calculations
| Radius | Central Angle | Chord Length | Application |
|---|---|---|---|
| 10 | 60° | 10.00 | Side of inscribed hexagon |
| 10 | 90° | 14.14 | Side of inscribed square |
| 10 | 120° | 17.32 | Side of equilateral triangle |
| 10 | 180° | 20.00 | Diameter (longest chord) |
Circle Segments and Related Measures
| Term | Definition | Formula | Relationship to Chord |
|---|---|---|---|
| Chord (c) | Straight line connecting two circle points | c = 2r·sin(θ/2) | Primary measure |
| Sagitta (h) | Height of circular segment | h = r - √(r² - (c/2)²) | Perpendicular to chord midpoint |
| Arc Length (s) | Length of circle circumference between chord endpoints | s = rθ | Always longer than chord |
| Segment Area | Area between chord and arc | A = ½r²(θ - sinθ) | Depends on chord and radius |
Real-World Applications
Architecture & Construction
- Arch design: Calculating chord lengths for circular arches and domes
- Bridge construction: Determining support placements for arched bridges
- Dome structures: Calculating rib lengths for geodesic domes
- Circular windows: Determining frame dimensions for round windows
Engineering & Manufacturing
- Gear design: Calculating tooth spacing on circular gears
- Pulley systems: Determining belt lengths for pulley arrangements
- Pipe fabrication: Calculating cut lengths for pipe intersections
- Metal working: Bending calculations for circular metal forms
Science & Mathematics
- Trigonometry: Fundamental in trigonometric identities and proofs
- Geometry: Essential for circle theorems and properties
- Physics: Circular motion and orbital calculations
- Astronomy: Calculating distances in circular orbits
Sports & Recreation
- Track design: Calculating lane markings on circular tracks
- Basketball court: Three-point line arc calculations
- Soccer field: Center circle and penalty arc markings
- Pool/billiards: Bank shot angles on circular tables
Step-by-Step Calculation Examples
Example 1: Radius = 10, Central Angle = 90°
- Convert angle to radians: 90° × π/180 = π/2 ≈ 1.5708 rad
- Divide by 2: θ/2 = 1.5708/2 = 0.7854 rad
- Calculate sine: sin(0.7854) = 0.7071
- Apply formula: c = 2 × 10 × 0.7071 = 14.142
- Result: Chord length = 14.142 units
- Verification: This is side of square inscribed in circle of radius 10
Example 2: Radius = 10, Distance from Center = 6
- Square radius: r² = 10² = 100
- Square distance: d² = 6² = 36
- Subtract: r² - d² = 100 - 36 = 64
- Take square root: √64 = 8
- Multiply by 2: c = 2 × 8 = 16
- Result: Chord length = 16 units
- Verification: Half-chord = 8, forms 6-8-10 right triangle with radius
Related Calculators
Frequently Asked Questions (FAQs)
Q: What's the difference between chord length and arc length?
A: Chord length is the straight-line distance between two points on a circle. Arc length is the distance along the curved circumference between those same two points. For any given two points on a circle, the chord length is always shorter than the arc length (except when the chord is a diameter of a semicircle).
Q: How do I find chord length without knowing the angle?
A: You can use the perpendicular distance method: c = 2√(r² - d²), where d is the perpendicular distance from the circle's center to the chord. Alternatively, if you know the sagitta (height of the segment), you can use: c = 2√(2rh - h²).
Q: What is sagitta and how is it related to chord length?
A: Sagitta (also called versine) is the height of the circular segment - the perpendicular distance from the chord's midpoint to the arc. The relationship is: h = r - √(r² - (c/2)²), or rearranged: c = 2√(2rh - h²). Sagitta is maximum when chord is small and minimum (zero) when chord is diameter.
Q: Can chord length be greater than diameter?
A: No, chord length cannot exceed the diameter. The diameter (2r) is the longest possible chord in a circle. All other chords are shorter. When central angle approaches 180°, chord length approaches diameter. When angle = 180°, chord = diameter.
Accurately calculate chord lengths with Toolivaa's free Chord Length Calculator, and explore more geometric tools in our Geometry Calculators collection.