circle chord calculator

What is a chord in a circle?

A chord is a straight line segment that connects two points on a circle. The diameter is a special chord that passes through the center and is the longest possible chord. In geometry, engineering, and design, chord calculations help convert curved geometry into practical straight-line measurements.

If you know the circle radius and one additional measurement (such as angle, sagitta, or center distance), you can compute the chord quickly and accurately.

How this circle chord calculator works

This tool supports three common ways to find chord length:

• Radius + central angle (most direct in trigonometry)
• Radius + distance from center to chord (common in CAD and fabrication)
• Radius + sagitta (common in arching, road geometry, and curved panel work)

In addition to chord length, the calculator also returns central angle, sagitta, arc length, and segment area for a complete geometric snapshot.

Core formulas

• Chord from angle: c = 2r sin(θ/2)
• Chord from center distance: c = 2√(r² - d²)
• Chord from sagitta: c = 2√(2rs - s²)
• Angle from center distance: θ = 2 arccos(d/r)
• Sagitta: s = r - d
• Arc length: L = rθ (θ in radians)

Why chord calculations matter

1) Construction and fabrication

When cutting curved parts from flat material, builders often need the straight-line span between arc endpoints. That span is the chord. Accurate chord values reduce fitting errors and material waste.

2) CAD, CNC, and layout work

Digital workflows frequently represent curves using arc definitions while machining paths may require discrete points or straight segments. Chord values help bridge these representations.

3) Transportation and civil design

In road and rail geometry, sagitta and chord relationships are used in curve staking, alignment checks, and geometric control.

Worked examples

Example A: Radius and angle known

Suppose radius r = 10 and central angle θ = 60°.

Chord: c = 2(10)sin(30°) = 20 × 0.5 = 10.

Example B: Radius and center distance known

Let radius r = 12 and center distance d = 9.

Chord: c = 2√(12² − 9²) = 2√(144 − 81) = 2√63 ≈ 15.875.

Example C: Radius and sagitta known

Let radius r = 20 and sagitta s = 3.

Chord: c = 2√(2(20)(3) − 3²) = 2√111 ≈ 21.071.

Common mistakes to avoid

• Mixing units (for example, radius in meters but sagitta in millimeters).
• Using degrees where radians are required in manual calculations of arc length and trig functions.
• Invalid geometry inputs such as center distance greater than or equal to radius for a real non-degenerate chord.
• Confusing sagitta with arc length—they are different measurements.

Quick interpretation guide

As central angle increases from small values toward 180°, the chord grows. At 180°, chord equals diameter (2r). For tiny angles, chord and arc length become very close. This behavior is useful when approximating gentle curves with straight segments.

Final notes

Use this calculator whenever you need a reliable chord measurement from basic circle inputs. It is especially useful for drafting, shop math, geometry classes, and practical field layout where speed and consistency matter.