Calculator Inputs
Use equations written as x² + y² + Dx + Ey + F = 0. This tool returns the center, radius, standard form, and a circle graph.
Graph and Export Tools
The chart plots the circle from the calculated center and radius. If r² is negative, the graph area shows the center only and notes the invalid radius.
Formula Used
Start from the general form of a circle:
Convert it by completing the square on x and y:
- Center: h = -D/2 and k = -E/2
- Radius squared: r² = (D² + E²)/4 - F
- Radius: r = √r², only when r² ≥ 0
How to Use This Calculator
- Enter the coefficient D from the x term.
- Enter the coefficient E from the y term.
- Enter the constant term F.
- Choose your preferred decimal precision.
- Enable full steps if you want the algebra shown.
- Press the convert button to see results above the form.
- Review the center, radius, standard form, and graph.
- Use the export buttons to save results as CSV or PDF.
Example Data Table
| General Form | Center | Radius | Standard Form |
|---|---|---|---|
| x² + y² - 6x + 8y - 11 = 0 | (3, -4) | 6 | (x - 3)² + (y + 4)² = 36 |
| x² + y² + 2x - 10y + 13 = 0 | (-1, 5) | √13 | (x + 1)² + (y - 5)² = 13 |
| x² + y² - 4x - 2y - 4 = 0 | (2, 1) | 3 | (x - 2)² + (y - 1)² = 9 |
| x² + y² + 8x + 6y + 9 = 0 | (-4, -3) | 4 | (x + 4)² + (y + 3)² = 16 |
Frequently Asked Questions
1. What is the general form of a circle?
The general form is x² + y² + Dx + Ey + F = 0. It combines squared terms, linear terms, and a constant. This calculator rewrites it into center-radius form.
2. What is the standard form of a circle?
The standard form is (x - h)² + (y - k)² = r². It directly shows the center coordinates and radius, making graphing and interpretation much easier.
3. How does the calculator find the center?
It uses h = -D/2 and k = -E/2. These values come from completing the square on the x and y groups in the equation.
4. How is the radius calculated?
The tool computes r² = (D² + E²)/4 - F. If that value is nonnegative, the radius is the square root of r². Negative values mean no real circle exists.
5. What happens when r² is negative?
A negative r² means the equation does not represent a real circle in the coordinate plane. The calculator still shows the transformed expression and warns about the invalid radius.
6. Can I use decimal coefficients?
Yes. The inputs accept integers and decimals. You can also control the output precision, which helps when classroom problems or design checks require rounded answers.
7. Why is completing the square important here?
Completing the square converts the equation from expanded form into geometric form. That step reveals the center and radius directly, which is why it is central to circle analysis.
8. What does the graph show?
The graph plots the calculated circle and marks the center. It helps you verify whether the algebraic conversion matches the expected geometry from the equation.