Find center and radius from x² + y² + Dx + Ey + F = 0
Result
Center and Radius
Step-by-Step Derivation
Circle Center and Radius Formula
center = (-D/2, -E/2), r² = (D/2)² + (E/2)² - F
The general circle equation can be converted to standard form by completing the square. The x coefficient D and y coefficient E determine the center, while the constant F helps determine radius squared.
⚠This tool assumes the x² and y² coefficients are both 1. If radius squared is negative, there is no real circle.
How Center and Radius Work
Completing the square rewrites the general equation into standard circle form, making the center and radius visible.
General Form
Use x²+y²+Dx+Ey+F=0.
Center
The center is (-D/2,-E/2).
Radius Squared
Compute r² before taking the square root.
Real Circle
r² must be zero or positive.
💡 Example: x²+y²-4x+6y-12=0 gives center (2,-3), r=5.
Applications of Circle Center and Radius
Circle GraphingCompleting the SquareAnalytic GeometryCoordinate Proofs
Frequently Asked Questions
What is a circle center radius calculator?▼
It finds the center and radius from the general equation x²+y²+Dx+Ey+F=0.
What is the formula for center and radius?▼
For x²+y²+Dx+Ey+F=0, center is (-D/2,-E/2) and r²=(D/2)²+(E/2)²-F.
How do I use this calculator?▼
Enter D, E, and F from the circle general form, then click Calculate.
What if radius squared is negative?▼
Then the equation has no real circle because the radius would be imaginary.
Where is this calculator useful?▼
It is useful for graphing circles, completing the square, and analytic geometry problems.
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