Circle Equation Calculator
A circle given in general form x² + y² + Dx + Ey + F = 0 is rewritten into standard form (x − h)² + (y − k)² = r² by completing the square on the x and y terms.
Enter Values
How to use this calculator
- Write your circle in general form x² + y² + Dx + Ey + F = 0, then enter the coefficients D, E and F (use the ± toggle for negatives).
- Read the centre (h, k) and radius r, along with the completed standard form (x − h)² + (y − k)² = r².
- Use the diameter, circumference and area outputs, or open the working panel to see each step of completing the square.
How it works
A circle given in general form x² + y² + Dx + Ey + F = 0 is rewritten into standard form (x − h)² + (y − k)² = r² by completing the square on the x and y terms. This gives the centre h = −D ÷ 2 and k = −E ÷ 2, and the radius from r² = h² + k² − F.
Once r is known, the derived quantities follow directly: diameter = 2r, circumference = 2πr, and area = πr². If h² + k² − F is negative the equation has no real circle (imaginary radius), and if it is exactly zero the equation collapses to a single point — the calculator flags both cases instead of returning a meaningless number.
Worked example
x² + y² − 4x − 6y − 12 = 0. With D = −4, E = −6, F = −12: h = −(−4)/2 = 2 and k = −(−6)/2 = 3, so r² = 2² + 3² − (−12) = 4 + 9 + 12 = 25 and r = 5. The circle is (x − 2)² + (y − 3)² = 25, centred at (2, 3) with radius 5, diameter 10, circumference 31.415927 and area 78.539816.
Common mistakes
- Forgetting the minus sign: the centre is (−D/2, −E/2), not (D/2, E/2), so a −4x term gives a centre x of +2.
- Confusing r² with r — the right-hand side of the standard form is the radius squared, so you must take the square root to get the actual radius.
- Entering coefficients for an equation that is not already in the form x² + y² + Dx + Ey + F = 0. If the x² and y² terms have a coefficient other than 1, divide the whole equation by that coefficient first.
Frequently asked questions
What form does this calculator expect?
The general form x² + y² + Dx + Ey + F = 0, where the x² and y² coefficients are both 1. If your equation has a common coefficient a on the squared terms (a·x² + a·y² + …), divide every term by a first.
How do I get the centre and radius from D, E and F?
The centre is (h, k) = (−D/2, −E/2) and the radius is r = √(h² + k² − F). These come from completing the square on the x and y terms.
Why does it sometimes say there is no real circle?
If h² + k² − F is negative, the radius squared is negative, so no real circle exists. If it equals exactly zero, the equation describes a single point rather than a circle, and the calculator reports that instead.
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