How to Use Chord Length Calculator
The Chord Length Calculator computes the length of a chord inside a circle using one of two input methods. Select the method that matches the information you have, fill in the values, and the result appears immediately.
- Choose input method — select “Radius + Central Angle” if you know the arc’s angle, or “Radius + Distance” if you know the perpendicular distance from the center to the chord.
- Enter the radius — type the circle’s radius in any consistent unit (meters, centimeters, inches, etc.).
- Enter the second value — type the central angle (in degrees or radians) or the perpendicular distance from the center.
- Read the result — the Chord Length Calculator displays the chord length and the substituted formula.
Make sure the distance d is not greater than the radius r; otherwise, no chord exists for that combination.
Formula & Theory - Chord Length Calculator
The Chord Length Calculator applies two equivalent geometric formulas:
Method 1 — Central angle:
c = 2r · sin(θ / 2)
Method 2 — Perpendicular distance:
c = 2 · √(r² − d²)| Symbol | Meaning |
|---|---|
| c | Chord length |
| r | Radius of the circle |
| θ | Central angle subtended by the chord |
| d | Perpendicular distance from the center to the chord |
Both formulas are derived from basic right-triangle geometry. In Method 1, the half-chord forms the opposite side of a right triangle with hypotenuse r and half-angle θ/2. In Method 2, the perpendicular from the center to the chord bisects the chord, so the half-chord length equals √(r² − d²).
Assumptions and Limits
The radius must be positive. For Method 1, the central angle must be between 0° and 360° (0 and 2π radians). For Method 2, the distance d must be non-negative and no larger than the radius r.
Use Cases for Chord Length Calculator
The Chord Length Calculator is useful in many practical and academic settings. Common uses include:
- Geometry homework — Find the chord length when given a circle diagram with a labeled central angle or a perpendicular distance.
- Engineering and drafting — Calculate chord lengths for circular arches, pipe sections, or cam profiles in mechanical design.
- Surveying — Determine the straight-line distance between two points on a circular boundary given the radius and the arc’s central angle.
- Architecture — Compute the span of an arch or curved window opening for structural calculations.
The Chord Length Calculator also displays the full substituted formula so you can verify each step of the calculation and understand exactly how the result was obtained.