Chord Length Calculator

What is the Chord of a Circle?

In Euclidean geometry, a chord is a straight line segment joining any two distinct points located on the circumference of a circle. Unlike an arc, which curves along the perimeter path, a chord cuts directly across the interior planar disk. The longest possible chord in any circle is the diameter ($d = 2r$), which passes directly through the circle center point. Every other chord lies strictly off-center and is strictly shorter than the diameter.

Chords serve as the mechanical and structural link between angular rotation, arc trajectories, and linear spatial distance. Whether determining spacing between holes in a flange using a bolt circle chord calculation, engineering roof trusses to size a top chord member, or analyzing the aerodynamic chord line of an aircraft wing profile, chord geometry translates circular coordinates into straight-line measurements.

Core Chord Length Formulas

Depending on which initial dimensions are provided, choose the matching geometric formula below to calculate chord length ($c$):

How to Find Chord Length of a Circle

A chord forms the base of an isosceles triangle whose two identical sides are formed by radii extending from the center of the circle. Dropping a perpendicular line (an apothem) from the center to the chord bisects both the chord and the central angle, creating two congruent right-angled triangles:

How to Find Radius from Chord Length and Arc Height (Sagitta)

In machine maintenance, tooling replacement, and archaeological reconstruction, measuring the radius directly is often impossible because the geometric center point is located in empty space. However, if you measure the straight chord span ($c$) across two points on the curved rim and the perpendicular height ($h$, the sagitta) from the chord center to the arc summit, you can isolate the radius using the intersecting chords theorem:

Specialized Chords: Bolt Circles, Roof Trusses, and Wing Aerodynamics

Outside standard circle geometry, the term chord defines critical linear load and reference lengths across multiple engineering fields:

How to Use This Chord Length Calculator Step-by-Step

This tool calculates straight chord lines, arc lengths, sagittas, and bolt hole centers instantly:

1. Select Input Mode: Choose your known parameters: Radius and Central Angle, Radius and Arc Length, or Radius and Sagitta Height.
2. Enter Numerical Dimensions: Type your measurements into the input boxes. Confirm that your angle mode (Degrees or Radians) matches your drawing specifications.
3. Review the Instant Solution: In real time (0ms delay), the calculation engine outputs:
   • The straight linear chord length ($c$).
   • The corresponding curved arc length ($s = r\theta$).
   • The apothem center distance ($d_{\text{center}}$) and sagitta height ($h = r - d_{\text{center}}$).

Quick Reference: Chord Lengths Across Common Radii and Angles

This reference table compares straight chord lengths against curved arc distances for a standardized circle with radius $r = 10\text{ units}$:

Worked Practical Examples with Step-by-Step Arithmetic

Practical Applications Across Industries

Common Pitfalls & Mistakes to Avoid

• Forgetting to Bisect the Angle ($\theta$ vs. $\theta/2$): The formula requires $\sin(\theta / 2)$, not $\sin(\theta)$. Forgetting to divide the central angle by 2 produces an incorrect chord value.
• Calculator Angle Mode Misconfiguration (Deg vs. Rad): Entering a $60^\circ$ angle when your scientific calculator or spreadsheet (like Microsoft Excel) is set to Radians produces an entirely incorrect result. Remember: Excel trigonometric functions like =SIN() always expect radians.
• Assuming Arc Length Equals Chord Length: A chord is always strictly shorter than its corresponding outer circular arc ($c < s$) for all angles greater than zero. They only converge to parity as $\theta$ approaches zero.
• Confusing Bolt Circle Radius with Pitch Diameter (BCD): In bolt circle drill templates, $\text{BCD}$ represents the full diameter across the hole pattern. Using radius instead of diameter will halve the required center-to-center drilling distance.