Circle Equation Calculator
From a circle’s centre (h, k) and radius, get its standard equation, general form, area and circumference.
How it works
Standard: (x − h)² + (y − k)² = r² · General: x² + y² + Dx + Ey + F = 0 with D = −2h, E = −2k, F = h² + k² − r² · Area = πr² · Circumference = 2πr
A circle is every point a fixed distance r from a centre (h, k), and squaring that distance condition gives the standard equation (x − h)² + (y − k)² = r². Expanding the two brackets and collecting terms produces the general form x² + y² + Dx + Ey + F = 0, where D = −2h, E = −2k and F = h² + k² − r². Standard form shows the centre and radius at a glance; general form is what you usually meet in coordinate-geometry problems and must complete the square to interpret.
Worked example
Centre (2, −3), radius 5. Standard form: (x − 2)² + (y + 3)² = 25 — note (y − (−3)) becomes (y + 3), and 5² = 25. Expanding gives x² + y² − 4x + 6y − 12 = 0, since D = −2·2 = −4, E = −2·(−3) = 6 and F = 2² + (−3)² − 5² = 4 + 9 − 25 = −12. The area is π·5² ≈ 78.5398 and the circumference is 2π·5 ≈ 31.4159.
Sources: Circle — Wikipedia
Frequently asked questions
What is the difference between standard and general form?
Standard form (x − h)² + (y − k)² = r² displays the centre (h, k) and radius r directly. General form x² + y² + Dx + Ey + F = 0 hides them inside the coefficients; you complete the square to recover the centre and radius from D, E and F.
Why does the centre’s sign flip in the equation?
The formula subtracts the centre: (x − h). A centre at h = 2 gives (x − 2), but a centre at k = −3 gives (y − (−3)) = (y + 3). So a negative coordinate appears as a plus sign inside the bracket.
How do I get the radius from the general form?
From x² + y² + Dx + Ey + F = 0, the centre is (−D/2, −E/2) and the radius is √((D/2)² + (E/2)² − F). This must be positive for a real circle; a zero or negative value means a point or no real circle.
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This calculator is for educational purposes. Double-check important results before acting on them.