What Is a Circle Equation?
A circle equation describes every point (x, y) that lies the same distance from a fixed point called the center. That constant distance is the radius. In coordinate geometry, circles show up in graphing, engineering drawings, navigation, design, physics, and any situation where you model “all points at a fixed distance.”
This circle equation calculator helps you work with the two most common forms: the standard (center-radius) form and the general form. It converts between them, shows the algebra steps (including completing the square), and also outputs useful circle measures such as diameter, circumference, and area.
Standard Form: The Fastest Way to Read Center and Radius
The standard form of a circle is: (x − h)² + (y − k)² = r². It’s often called center-radius form because it directly reveals the center (h, k) and the radius r. If you are graphing or checking circle properties, this is usually the most convenient form.
For example, (x − 2)² + (y + 1)² = 9 has center (2, −1) and radius 3 because r² = 9. Notice how the signs work: (y + 1)² means k = −1 because y − (−1) = y + 1.
General Form: The Form You Often See After Expanding
When you expand the squared terms, many circle equations appear as: x² + y² + Dx + Ey + F = 0. This is called the general form. It is common in algebra problems, coordinate geometry exercises, and when equations come from manipulation of other expressions.
General form doesn’t show the center and radius immediately, but you can convert it to standard form by completing the square in x and y. This calculator does that conversion for you and shows the intermediate steps so you can follow the logic.
How to Convert General Form to Standard Form by Completing the Square
To convert x² + y² + Dx + Ey + F = 0, group x terms and y terms: (x² + Dx) + (y² + Ey) = −F. Then complete the square in each group: x² + Dx becomes (x + D/2)² − (D/2)², and y² + Ey becomes (y + E/2)² − (E/2)².
After rearranging, you’ll get (x − h)² + (y − k)² = r², where the center is (−D/2, −E/2) and r² is the constant on the right side once all terms are moved appropriately.
How to Convert Standard Form to General Form
Going from standard to general is mostly expansion. Start with (x − h)² + (y − k)² = r², expand both squares, combine like terms, then move everything to one side to reach x² + y² + Dx + Ey + F = 0. This is helpful if a problem expects a final answer in general form.
The calculator will show the expanded coefficients and the final general equation so you can copy it quickly into homework, notes, or graphing tools.
Center, Radius, Diameter, Circumference, and Area
Once you know the radius, many other circle values follow:
- Diameter: 2r
- Circumference: 2πr
- Area: πr²
These outputs are useful for planning, geometry checks, and quick validation. For example, if a conversion produces an unexpectedly large area, that can hint at a coefficient/sign mistake in the original equation.
When a Circle Equation Is Not a Real Circle
Not every equation that looks like a circle produces a real circle. After converting to standard form, you should check r². If r² is negative, the radius would be the square root of a negative number, which is not a real value. In the real coordinate plane, that means the equation does not represent a circle you can draw.
This calculator calls that out clearly by showing r² and indicating whether a real radius exists.
Common Mistakes and How to Avoid Them
Many circle-equation mistakes come from sign handling and coefficient transfer. Keep these checks in mind:
- In (x − h)², the center x-coordinate is h. In (x + 3)², h = −3.
- In general form, the center is (−D/2, −E/2), not (D/2, E/2).
- If you expand, combine carefully: −2hx and −2ky are where D and E come from.
- Always check if r² is non-negative before assuming a real circle.
Examples You Can Try Right Now
If you want a quick confidence check, try these:
- Standard: h = 2, k = −1, r = 3 → (x − 2)² + (y + 1)² = 9
- General: D = −4, E = 2, F = −4 → x² + y² − 4x + 2y − 4 = 0
- Center at origin: (x)² + (y)² = 16 → radius 4
- No real circle example: x² + y² + 2x + 2y + 10 = 0 (will produce negative r²)
Limitations and Safe Use Notes
This tool performs numeric conversions and shows algebra steps in a clear, instructional way. If your inputs are fractions or exact symbolic expressions, results will still be displayed numerically using your chosen precision. For advanced symbolic simplification, a computer algebra system may be required.
If you’re using circle equations for real-world work (CAD, engineering, surveying), verify units and coordinate conventions and double-check the equation source before using results in final decisions.
FAQ
Circle Equation Calculator – Frequently Asked Questions
Answers about standard vs general form, completing the square, center/radius, and when an equation is not a real circle.
The standard form is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius.
A common general form is x² + y² + Dx + Ey + F = 0. You can convert it to standard form by completing the square in x and y.
For x² + y² + Dx + Ey + F = 0, the center is (−D/2, −E/2) and the radius is √((D/2)² + (E/2)² − F), assuming the value under the square root is non-negative.
Completing the square rewrites expressions like x² + Dx into (x + D/2)² − (D/2)². It helps convert the general circle equation into center-radius form.
Yes. If r² is negative after converting to standard form, the equation does not represent a real circle in the coordinate plane.
Center-radius form is another name for the standard form (x − h)² + (y − k)² = r², which directly shows the center (h, k) and radius r.
Expand (x − h)² + (y − k)² = r² to get x² + y² − 2hx − 2ky + (h² + k² − r²) = 0, which matches x² + y² + Dx + Ey + F = 0.
Yes. It shows how the equation is rearranged, how completing the square is applied, and how the center and radius are derived.
Yes. Once the radius is known, the calculator returns diameter (2r), circumference (2πr), and area (πr²).
Yes. The inputs and results are responsive and work on phones, tablets, and desktops.