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A circle's standard-form equation, (x−h)²+(y−k)²=r², directly encodes its center (h,k) and radius r — it comes straight from the distance formula, since it says every point (x,y) on the circle is exactly r units from the center (h,k). This calculator converts in both directions — from a known center and radius to the equation, or from a messier general-form equation (x²+y²+Dx+Ey+F=0, the form you'll often get after expanding or from a textbook problem) back to the center and radius, by completing the square on both the x-terms and y-terms.
How it works
Choose a direction. Given the center (h,k) and radius r, the calculator writes the standard-form equation directly by substituting into (x−h)²+(y−k)²=r². Given general-form coefficients from x²+y²+Dx+Ey+F=0, it completes the square on the x-terms and y-terms separately: grouping x²+Dx into (x+D/2)²−(D/2)² and y²+Ey into (y+E/2)²−(E/2)², then moving the constants to the other side. This reveals the center as h = −D/2, k = −E/2, and the radius as r = √(h²+k²−F). If h²+k²−F works out negative, the equation doesn't describe any real circle at all.
- Enter direction.
- Enter center h.
- Enter center k.
- Enter radius (r).
- Enter d (coefficient of x).
- Enter e (coefficient of y).
- Enter f (constant).
- Click Calculate to see your results.
Examples
Center (2, 3), radius 5
Standard form: (x−2)² + (y−3)² = 25.
General form x² + y² − 4x + 6y − 3 = 0
Here D=−4, E=6, F=−3, so h = −(−4)/2 = 2, k = −6/2 = −3, and r² = 2² + (−3)² − (−3) = 4 + 9 + 3 = 16, giving r = 4. Standard form: (x−2)² + (y+3)² = 16, center (2, −3).
Center (0, 0), radius 7
Standard form: x² + y² = 49 — a circle centered at the origin simplifies since h = k = 0.
Who should use it
- Converting a textbook general-form circle equation into an easily-graphable standard form.
- Writing a circle's equation from a known center and radius for a geometry or algebra problem.
- Checking whether a given quadratic equation actually represents a real circle.
Industry applications
- Algebra and analytic geometry education
- Computer graphics and CAD (circle definitions)
Advantages
- Converts in both directions — center/radius to equation, and equation to center/radius.
- Shows the completing-the-square work so the conversion isn't a black box.
Limitations
- Only handles equations of an actual circle — an ellipse or other conic section needs a different approach entirely.
Common mistakes to avoid
- Forgetting to negate D and E when finding h and k from the general form — the correct relationship is h = −D/2, not h = D/2.
- Skipping the check for a negative r² result, which signals no real circle exists.
- Misreading a "+3" inside the parentheses as h = 3 instead of h = −3 — the standard form always subtracts h and k, so a plus sign means the center coordinate itself is negative.
Best practices
- Always verify your final answer by expanding the standard form back out and checking it matches the original general-form equation.
- Double check the sign convention when reading h and k directly off a standard-form equation containing a "+" instead of a "−".
- Watch for a negative r² result as a signal that the given equation has no real circle solution.
Tips
- Working with a square inscribed in a circle instead? See the Square in a Circle Calculator.