Library / Geometry, coordinates, surveying and trigonometry / GEO-39
Geometry, coordinates, surveying and trigonometry
Common chord of two intersecting circles
Length of the chord shared by two circles that cross, from the two radii and the distance between centres.
- Ref
- GEO-39
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- c [cm]
Two circles of radii R and r whose centres are d apart cross when |R − r| ≤ d ≤ R + r. The common chord is perpendicular to the line of centres. The distance from the first centre to the chord is x = (d² + R² − r²)/(2d), and the half-chord is √(R² − x²), so c = 2√(R² − x²). At tangency the chord is zero. Concentric circles are excluded.
Length of the chord shared by two circles that cross, from the two radii and the distance between centres. At tangency it is zero, and concentric circles are excluded.
Calculator
- Distance from the first centre to the chord3 cm
- Distance from the second centre to the chord3 cm
- Distance between the centres at which the circles just touch externally10 cm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Circles of radius 5 and 5 cm with centres 6 cm apart share a chord of 8 cm.
- The standard 5, 5 and 6 example gives 8 cm, since the half-chord, the 3 cm half distance and the 5 cm radius make a 3–4–5 triangle.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| d Distance between the centres | 1 to 10 cm | 0 | 9.95 | 9.95 | +10% | −6.09 % |
| R Radius of circle 1 | 1.5 to 10.5 cm | 2.031 | 10 | 7.969 | +10% | +7.34 % |
| r Radius of circle 2 | 1.5 to 10.5 cm | 2.031 | 10 | 7.969 | +10% | +7.34 % |
Sweep of Distance between the centres
| Distance between the centres (cm) | Common chord length (cm) |
|---|---|
| 1 | 9.95 |
| 2.125 | 9.772 |
| 3.25 | 9.457 |
| 4.375 | 8.992 |
| 5.5 | 8.352 |
| 6.625 | 7.491 |
| 7.75 | 6.32 |
| 8.875 | 4.608 |
| 10 | 0 |
Sweep one variable
Variables and how each one contributes
Common chord length
What the equation gives you. Everything else feeds it.
Radius of circle 1
The radius of the first circle, in cm. A larger circle lengthens the chord, as long as the two still cross. Raising it by 10 % (to 5.5) moves the result by +7.34 %, from 8 to 8.587 cm.
Radius of circle 2
The radius of the second circle, in cm. A larger circle lengthens the chord, as long as the two still cross. Raising it by 10 % (to 5.5) moves the result by +7.34 %, from 8 to 8.587 cm.
Distance between the centres
The distance between the two centres, in cm. Moving the circles apart shortens the chord, which falls to zero at tangency. Raising it by 10 % (to 6.6) moves the result by −6.09 %, from 8 to 7.513 cm.
Worked example with the values above
- Distance from the first centre to the chord: x = (36 + 25 − 25) / (2 × 6) = 3 cm.
- Half chord: √(R² − x²) = √(25 − 9) = 4 cm.
- c = 8 cm.
Assumptions and limits
- Plane Cartesian coordinates, and a circle of positive radius unless stated.
- Angles in radians measured anticlockwise from the positive x-axis.
Common mistakes
- Using circles that do not cross, where the radicand is negative.
- Mixing up which radius goes with which centre in the distance formula.
- Reading the chord as the distance between the centres.
Statistics and uncertainty
These are exact results from the circle and line equations, but they need a root of a difference. Where the radicand is close to zero, as near tangency, small input errors can change the answer a lot or change whether a solution exists at all.
Where it is used
- Circle fitting, wheel and cam geometry, hole patterns and clearance checks.
- Finding where a path, ray or cut meets a circular boundary.
Maths you need
- Equation of a circle: (x − h)² + (y − k)² = r²
- Quadratic discriminant: b² − 4ac decides the number of roots
- Gradient of a normal: perpendicular to the tangent