Library / Geometry, coordinates, surveying and trigonometry / GEO-36
Geometry, coordinates, surveying and trigonometry
Centre and radius of a circle by completing the square
Radius of the circle x² + y² + Dx + Ey + F = 0.
- Ref
- GEO-36
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- r [cm]
The circle x² + y² + Dx + Ey + F = 0 can be rewritten as (x + D/2)² + (y + E/2)² = (D² + E²)/4 − F by completing the square. So the centre is (−D/2, −E/2) and the radius is √[(D² + E²)/4 − F]. If the term under the root is zero the locus is a single point, and if it is negative there are no real points.
Radius of the circle x² + y² + Dx + Ey + F = 0. The centre is (−D/2, −E/2), in the derived rows. A positive radicand gives a circle, zero gives a single point and a negative value gives no real locus.
Calculator
- Centre x, −D/23 cm
- Centre y, −E/2−2 cm
- Diameter10 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.
- x² + y² − 6x + 4y − 12 = 0 is a circle of radius 5 cm centred at (3, −2).
- For F ≥ (D² + E²)/4 = 13 the circle shrinks to a point and then disappears.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| D Coefficient of x | −20 to 20 cm | 4 | 10.77 | 6.77 | +0.8 cm | −4.59 % |
| E Coefficient of y | −20 to 20 cm | 4.583 | 11 | 6.417 | +0.8 cm | +3.46 % |
| F Constant | −30 to 12 cm² | 1 | 6.557 | 5.557 | +0.84 cm² | −1.69 % |
Sweep of Constant
| Constant (cm²) | Radius (cm) |
|---|---|
| −30 | 6.557 |
| −24.75 | 6.144 |
| −19.5 | 5.701 |
| −14.25 | 5.22 |
| −9 | 4.69 |
| −3.75 | 4.093 |
| 1.5 | 3.391 |
| 6.75 | 2.5 |
| 12 | 1 |
Sweep one variable
Variables and how each one contributes
Radius
What the equation gives you. Everything else feeds it.
Coefficient of x
The coefficient of x in the equation, in cm. Moves the centre in x and enlarges the radius as D grows, since it enters squared. Raising it by 0.5 (to −5.5) moves the result by −2.92 %, from 5 to 4.854 cm.
Coefficient of y
The coefficient of y in the equation, in cm. Moves the centre in y and enlarges the radius as E grows, since it enters squared. Raising it by 0.5 (to 4.5) moves the result by +2.1 %, from 5 to 5.105 cm.
Constant
The constant term, in cm². A larger F makes the circle smaller. Shrinks the circle, and reduces the radius to zero at F = (D² + E²)/4. Raising it by 0.5 (to −11.5) moves the result by −1.01 %, from 5 to 4.95 cm.
Worked example with the values above
- Centre: (−D/2, −E/2) = (3, −2).
- Radicand: (D² + E²)/4 − F = (36 + 16)/4 − (−12) = 25.
- r = √25 = 5 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
- Forgetting the signs of D and E when reading the centre: it is (−D/2, −E/2).
- Not checking that the radicand is positive before taking the root.
- Reading an equation with x² and y² having different coefficients as a circle without dividing through first.
Statistics and uncertainty
The radius comes from a difference of two numbers under a root, so when the radicand is small the radius is very sensitive to errors in D, E and F.
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