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Chord Length

A chord is a line segment whose endpoints lie on a circle. Length from angle: c = 2r·sin(θ/2). From distance d: c = 2√(r²−d²). Longest chord = diameter.

Concept Fundamentals
c = 2r·sin(θ/2)
From angle
c = 2√(r²−d²)
From distance
Longest chord (θ=180°)
Diameter
d=0 → c=2r
At center

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The diameter is the longest chord—when θ=180° or d=0. Chord length from distance uses the Pythagorean theorem on the right triangle. At d=r the chord length is 0 (point at the circle edge).

Key quantities
c = 2r·sin(θ/2)
From angle
Key relation
c = 2√(r²−d²)
From distance
Key relation
Longest chord (θ=180°)
Diameter
Key relation
d=0 → c=2r
At center
Key relation

Ready to run the numbers?

Why: Chords appear in architecture (arched doorways), music (string lengths), and GPS (circular coverage). The longest chord is the diameter.

How: From central angle: c = 2r·sin(θ/2). From perpendicular distance d: c = 2√(r²−d²) via Pythagorean theorem. Chords closer to center are longer.

The diameter is the longest chord—when θ=180° or d=0.Chord length from distance uses the Pythagorean theorem on the right triangle.
Sources:Wolfram MathWorld - ChordKhan Academy - Geometry

Run the calculator when you are ready.

Chord Length CalculatorEnter radius and central angle or distance from center
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GEOMETRYCircle Chord

Chord Length — c = 2r·sin(θ/2)

Calculate chord from radius + central angle OR perpendicular distance from center.

Arc Length →Segment Area →

📐 Real-World Examples — Click to Load

Calculation Settings

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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Chord length from angle: c = 2r·sin(θ/2).

— Formula

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The diameter is the longest chord of a circle.

— Property

📋 Key Takeaways

  • • A chord is a line segment whose endpoints lie on the circle
  • • The longest chord is the diameter (θ=180° or d=0)
  • • Chord length from angle: c = 2r·sin(θ/2) — θ in radians for sin
  • • Chord length from distance: c = 2√(r² - d²) — from Pythagorean theorem
  • • Chords closer to the center are longer; at d=r the chord length is 0

💡 Did You Know?

🎸Guitar frets are positioned using chord-length geometry — the 12th fret is exactly halfway along the stringSource: Music Theory
🌉Arch bridges use chord geometry to calculate the curved span and load distributionSource: Structural Engineering
🛢️Partially filled cylindrical tanks use chord length to calculate liquid volume from level heightSource: Chemical Engineering
📐The chord subtends an arc — chord length is always less than arc length for the same endpointsSource: Euclidean Geometry
🔬Particle accelerators use circular paths; chord calculations help design beam trajectoriesSource: Physics
🎯The perpendicular from center to a chord always bisects the chordSource: Circle Theorems

📖 Formulas Explained

From Central Angle

Using the isosceles triangle formed by two radii and the chord, sin(θ/2) = (c/2)/r, so c = 2r·sin(θ/2).

c=2r⋅sin⁡(θ2)c = 2r \cdot \sin\left(\frac{\theta}{2}\right)

From Distance

By Pythagorean theorem: r² = d² + (c/2)², so (c/2)² = r² - d², giving c = 2√(r² - d²).

c=2r2−d2c = 2\sqrt{r^2 - d^2}

🎯 Expert Tips

💡 Angle in Degrees

The calculator converts degrees to radians internally. Enter θ in degrees (0° to 360°).

💡 Distance Constraint

Distance d must satisfy 0 ≤ d < r. At d = r the chord degenerates to a point.

💡 Diameter Check

For θ = 180° or d = 0, chord = 2r (diameter). Use this to verify your inputs.

💡 Pipe Applications

For partially filled pipes, measure the liquid height h; then d = r - h.

⚖️ Comparison Table

FeatureThis CalculatorBasicManual
Angle & distance methods✅⚠️ Limited❌
Step-by-step solutions✅❌✅
Interactive charts✅❌❌
Copy & share✅❌❌
7 real-world examples✅❌❌

📊 Chord Quick Facts

2r
Max Chord (Diameter)
0
Min (d=r)
2
Input Methods
sin
Angle Formula

❓ FAQ

What is the longest chord?

The diameter — when θ = 180° or d = 0. Length = 2r.

Can chord length exceed the diameter?

No. The maximum chord length is the diameter (2r).

What if d equals the radius?

Then the chord length is 0 — the line from center touches the circle at one point (tangent).

Chord vs arc length?

Chord is the straight-line distance; arc is the curved distance. Arc ≥ chord.

How to find θ from chord and radius?

Use θ = 2·arcsin(c/(2r)), then convert radians to degrees.

How to find d from chord and radius?

Use d = √(r² - (c/2)²).

Units for angle?

Enter angle in degrees. The calculator converts to radians for the sine function.

When to use which formula?

Use angle when you know the central angle; use distance when you know how far the chord is from the center.

⚠️ Disclaimer: Results are based on Euclidean geometry. Real-world measurements may vary. For engineering applications, verify with appropriate standards.

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