Equation of a Circle Calculator: Standard & General Form Guide (2026)
Find the equation of a circle from center and radius, three points, or diameter endpoints. Convert between standard and general forms.
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Put these formulas into practice with our instant, step-by-step Equation of a Circle Calculator.
The equation of a circle is one of the most important formulas in coordinate geometry. Whether you are given the center and radius, three points on the circle, or diameter endpoints, our equation of a circle calculator finds the complete equation in both standard and general form. Understanding these two forms — and how to convert between them — is essential for solving problems in geometry, algebra, physics, and engineering.
Key Takeaway
The standard form (x - h)² + (y - k)² = r² directly reveals the center (h, k) and radius r. The general form x² + y² + Dx + Ey + F = 0 is useful for algebraic manipulation but requires completing the square to extract the center and radius.
Standard Form vs General Form
The standard form (x - h)² + (y - k)² = r² is the most intuitive representation. It tells you immediately that every point (x, y) on the circle is exactly r units from the center (h, k). The general form x² + y² + Dx + Ey + F = 0 expands the squared terms and groups like terms. While less intuitive, the general form is useful for solving systems of equations and finding intersections.
General: x² + y² + Dx + Ey + F = 0
Conversion: D = -2h, E = -2k, F = h² + k² - r²
How to Use the Calculator
The calculator supports three input methods:
- Center and Radius: Enter (h, k) and r to get the equation directly.
- Three Points: Enter three points on the circle. The calculator solves the system of equations to find the center and radius.
- Diameter Endpoints: Enter the two endpoints of a diameter. The center is the midpoint, and the radius is half the distance.
Worked Example
Find the equation of a circle with center (3, -2) and radius 5. Standard form: (x - 3)² + (y + 2)² = 25. Expanding: x² - 6x + 9 + y² + 4y + 4 = 25. General form: x² + y² - 6x + 4y - 12 = 0.
Frequently Asked Questions
- What is the standard form of a circle equation?
- (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
- How do I find the equation from three points?
- Substitute each point into x² + y² + Dx + Ey + F = 0 to get three equations. Solve for D, E, F.
- Can three collinear points define a circle?
- No. Three collinear points cannot form a circle; the system of equations becomes inconsistent.
Conclusion
The equation of a circle connects algebra and geometry in a powerful way. Use our free equation of a circle calculator to find standard and general form equations from any input.