Convert a circle equation from general form to standard form step by step. Shows completing the square process, center, radius, area, circumference, and detailed algebraic walkthrough.
Converting a circle equation from general form to standard form is one of the most important skills in analytic geometry and conic sections. The general form x² + y² + Dx + Ey + F = 0 is algebraically compact but geometrically opaque — you cannot read off the center or radius directly. The standard form (x − h)² + (y − k)² = r² immediately reveals the center at (h, k) and the radius r.
The conversion process relies on completing the square, a technique where you transform x² + Dx into a perfect square trinomial (x + D/2)² by adding (D/2)² to both sides, and similarly for y. This calculator walks you through every algebraic step so you can follow the logic and verify your own work.
After entering the coefficients D, E, and F, the tool instantly produces the standard form equation, center coordinates, radius, and derived quantities such as area, circumference, and diameter. It also highlights the completing-the-square contributions from the x and y terms with visual bars, helping you understand which term has a larger impact. A detailed step table shows each transformation from start to finish, with an option for a condensed summary view. Eight preset equations let you practice common examples without manual entry. This tool is ideal for algebra students studying conic sections, teachers preparing worked examples, and anyone who needs a quick, reliable conversion with a transparent derivation.
Use this calculator when you want to see the actual algebra behind a circle conversion instead of only the final answer. It is designed for situations where you need the standard form, center, and radius, but also want the completing-the-square steps broken out clearly enough to spot sign mistakes and understand where every term comes from.
That makes it especially helpful for students practicing conic sections, tutors preparing worked examples, and anyone reviewing older notes where the equation is still in general form. The combination of final outputs, intermediate square terms, and step tables turns it into both a solver and a teaching aid.
General form: x² + y² + Dx + Ey + F = 0. Complete the square: h = −D/2, k = −E/2, r² = (D/2)² + (E/2)² − F. Standard form: (x − h)² + (y − k)² = r².
Result: Standard form: (x − 3)^2 + (y + 2)^2 = 25
With D = -6, E = 4, and F = -12, the completing-the-square additions are (D/2)^2 = 9 and (E/2)^2 = 4. That gives center (3, -2), radius squared 25, and radius 5. The calculator rewrites the equation as (x − 3)^2 + (y + 2)^2 = 25 and then shows the matching area, circumference, and diameter.
A circle written as x² + y² + Dx + Ey + F = 0 is algebraically complete but not visually descriptive. Standard form, by contrast, immediately shows the center and radius. The bridge between the two is completing the square for the x-group and the y-group separately. Once those perfect squares are formed, the equation becomes much easier to interpret geometrically and to graph by hand.
Most errors happen in three places: halving the coefficients incorrectly, forgetting to square the halved values, or adding the square terms on one side without balancing the other side. Sign mistakes are also common when rewriting expressions like y² + 4y as (y + 2)² - 4. A step-by-step tool is useful because it keeps those transformations explicit and lets you compare your own algebra line by line against the correct structure.
Once the equation is in standard form, the radius is no longer just a symbol inside the equation. It becomes a usable measurement for area, circumference, diameter, and graphing scale. That is important in coordinate geometry because the same conversion often leads directly into tangent problems, distance checks, or sketching the circle on a grid. Seeing the standard form alongside the derived measurements helps connect the algebraic manipulation to the geometric meaning.
General form (x²+y²+Dx+Ey+F=0) is the expanded polynomial. Standard form ((x−h)²+(y−k)²=r²) explicitly shows the center and radius.
Completing the square rewrites x²+Dx as (x+D/2)²−(D/2)², creating a perfect square that reveals the center. Use this as a practical reminder before finalizing the result.
Divide the entire equation by the coefficient first so x² and y² both have coefficient 1, then enter D, E, F. Keep this note short and outcome-focused for reuse.
F = 0 means the origin lies on the circle (it satisfies 0²+0²+0+0+0=0). The formulas still work identically.
Calculations use standard 64-bit floating point. You can display up to 6 decimal places for precision.
r² = 0 means the equation represents a single point (degenerate circle). This happens when (D/2)² + (E/2)² = F.