The circle chord formula provides the exact length of a line segment connecting two points on a circle, based on radius and central angle. Understanding this relationship helps in geometry, engineering, and design problems where precise segment measurements matter.
This guide explains the core formula, practical uses, and common questions so you can apply chords in real calculations and diagrams.
| Term | Definition | Formula | Example with r = 5, θ = 60° |
|---|---|---|---|
| Chord | Segment joining two points on a circle | c = 2r sin(θ/2) | c ≈ 5.00 |
| Radius | Distance from center to circle edge | r = c / (2 sin(θ/2)) | r = 5 |
| Central Angle | Angle subtended by chord at circle center (degrees or radians) | θ = 2 arcsin(c / (2r)) | θ = 60° |
| Arc Length | Distance along circle between chord endpoints | s = r θ (θ in radians) | s ≈ 5.24 |
Circle Chord Formula Basics
The chord length depends on the radius and the central angle between the endpoints. The standard circle chord formula is c = 2r sin(θ/2), where c is chord length, r is radius, and θ is the central angle.
When θ is in radians, input directly into the formula; when in degrees, convert first or use a calculator that supports degree mode. This relationship remains consistent for any circle, making the formula broadly applicable.
Deriving the Formula from Trigonometry
To derive the formula, draw a line from the center to each endpoint, forming an isosceles triangle. Splitting this triangle in half creates a right triangle where sin(θ/2) = (c/2) / r.
Rearranging yields the circle chord formula c = 2r sin(θ/2). This derivation shows why the chord grows with larger radius and wider angles up to 180 degrees.
Using Chord Length in Real Problems
In engineering and architecture, the circle chord formula helps size curved elements like arches, bridges, and circular trusses. Knowing chord length allows precise material cuts and spacing.
Surveyors and astronomers also use chords to measure distances across curves and to relate angular separations to linear distances on lenses or celestial bodies.
Special Cases and Common Mistakes
When the central angle is 180°, the chord equals the diameter, giving c = 2r as the maximum chord length for a given circle. For very small angles, chord length approximates arc length closely.
Common mistakes include using degrees without switching calculator mode, confusing chord with arc length, or misidentifying the angle as an inscribed angle rather than a central angle.
Key Takeaways for Applying the Formula
- Use c = 2r sin(θ/2) to find chord length from radius and central angle.
- Verify calculator angle mode matches your unit for θ.
- Maximum chord occurs at θ = 180°, where it equals the diameter.
- Chord calculations are foundational for arcs, segments, and circular design.
FAQ
Reader questions
How do I calculate chord length if I only know the radius and arc length?
First convert arc length to central angle using θ = s / r, then apply c = 2r sin(θ/2) to find chord length.
Can the circle chord formula work in 3D space?
Yes, as long as the points lie on a circle, project them to the circle’s plane, find the radius and angle, then use c = 2r sin(θ/2).
What if the angle is given in degrees instead of radians?
Switch your calculator to degree mode or convert degrees to radians with θ_rad = θ_deg × π / 180 before using the formula.
How does chord length compare with arc length for the same angle?
Chord length is always shorter than or equal to arc length for the same radius and central angle, becoming equal only at zero angle.