chord length calculator

What Is a Chord in a Circle?

In circle geometry, a chord is a straight line segment whose endpoints both lie on the circle. The longest possible chord is the diameter, because it passes through the center and spans the circle at its widest point.

Chord length shows up in many practical areas: surveying, bridge design, arch layout, CNC machining, CAD drafting, and even game development when circular paths are approximated by straight segments.

How This Chord Length Calculator Works

This calculator gives you three ways to compute chord length depending on what measurements you already know.

1) Radius and Central Angle

When you know the radius r and the central angle θ, use:

c = 2r sin(θ/2)

This is often the easiest method in trigonometry classes and CAD systems where angle data is already available.

2) Radius and Distance from Center to Chord

If you know the radius r and the perpendicular distance from the center to the chord d, use:

c = 2√(r² − d²)

This comes directly from the Pythagorean theorem using half of the chord as one side of a right triangle.

3) Radius and Sagitta (Arc Height)

The sagitta h is the height from the chord to the arc. If r and h are known:

c = 2√(2rh − h²)

This method is useful in construction and fabrication when arc rise is easier to measure than an angle.

Step-by-Step Usage

• Select the method matching your known values.
• Enter radius and the second measurement.
• For the angle method, choose degrees or radians.
• Click Calculate Chord Length to get the result instantly.

Worked Example

Suppose a circle has radius 10 and central angle 60°. Then:

c = 2(10)sin(60°/2) = 20sin(30°) = 20(0.5) = 10

So the chord length is 10 units.

Common Mistakes to Avoid

• Mixing angle units: Using degrees in a radians-only formula causes wrong answers.
• Inconsistent units: Keep all linear values in the same unit system.
• Invalid geometry: In the distance method, d must be less than or equal to r.
• Premature rounding: Round only at the end for better precision.

Where Chord Length Is Used in Real Life

• Road and railway curve design
• Pipe and tank layout
• Architecture and arched openings
• CNC and laser path segmentation
• Computer graphics and simulation

Quick Geometry Notes

As the central angle increases from 0° to 180°, chord length increases from 0 to the diameter. For very small angles, the chord length becomes close to arc length, which is useful for approximations in engineering workflows.

If you frequently work with circles, also keep arc length, sector area, and sagitta formulas handy—they pair naturally with chord calculations.

Conclusion

This chord length calculator is built to be fast, flexible, and practical. Whether you know angle data, center offset, or sagitta, you can quickly compute accurate chord lengths and move on to your next design or math step with confidence.