Equation of a circle calculator

The equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r². It is the distance formula and nothing more: a point (x, y) lies on the circle exactly when its distance from the center is r, and squaring both sides of √((x - h)² + (y - k)²) = r removes the root. For center (2, -3) and radius 5 that gives (x - 2)² + (y + 3)² = 25.

The signs are where people slip. The formula subtracts the center, so a center coordinate of -3 becomes y - (-3) = y + 3. Read the other way, (x + 4)² means h = -4, not 4. And the right-hand side is r squared, not r: (x - 1)² + y² = 16 has radius 4, and (x - 1)² + y² = 7 has radius √7.

General form is the same equation multiplied out: x² + y² + Dx + Ey + F = 0. Expanding (x - h)² gives x² - 2hx + h², so D = -2h, E = -2k and F = h² + k² - r². For the example, D = -4, E = 6 and F = 4 + 9 - 25 = -12, so the general form is x² + y² - 4x + 6y - 12 = 0. When the center has fractional coordinates the coefficients do too, and multiplying the whole equation through by a common denominator gives whole-number coefficients; the page shows that version when it differs.

Going back from general form to the center and radius is completing the square. Group the x terms and the y terms and move the constant across: (x² - 4x) + (y² + 6y) = 12. Half of -4 is -2, and its square, 4, completes the first bracket; half of 6 is 3, and its square, 9, completes the second. Add both to each side and the brackets fold up: (x - 2)² + (y + 3)² = 25. If the squared terms share a coefficient other than 1, as in 2x² + 2y² - 8x + 12y - 24 = 0, divide everything by it first.

Not every equation of that shape is a circle. After completing the square, the right-hand side has to be positive. If it is zero, the only point that satisfies the equation is the center itself: a point circle, of radius 0. If it is negative, no real point satisfies it at all, because a sum of two squares cannot be negative; some texts call this an imaginary circle. The squared terms must also have equal coefficients and there must be no xy term. Unequal coefficients of the same sign describe an ellipse, opposite signs a hyperbola, and a missing square a parabola. The reverse mode reports which.

To check an answer, substitute a point you know is on the circle. (7, -3) is 5 to the right of the center, and 49 + 9 - 28 - 18 - 12 = 0.