Equation of a Circle Calculator
Enter a center point and radius, three points on the circle, or the coefficients of the general form to get the equation of the circle in standard form, general form, and parametric form. The calculator also shows the area, circumference, diameter, x- and y-intercepts, domain, range, and every step of the working.
Formula
Worked example
Circle with center (3, -2) and radius 5: standard form is (x - 3)^2 + (y + 2)^2 = 25. General form: D = -6, E = 4, F = -12, giving x^2 + y^2 - 6x + 4y - 12 = 0. Area = pi * 25 = 78.5398 units^2. Circumference = 2 * pi * 5 = 31.4159 units.
What is the equation of a circle?
A circle is the set of all points in a plane that are the same distance (the radius) from a fixed point (the center). If the center is at (h, k) and the radius is r, every point (x, y) on the circle satisfies the Pythagorean identity applied to the horizontal and vertical distances from the center: (x - h) squared plus (y - k) squared equals r squared. This is the standard form equation of the circle. Expanding and collecting terms gives the general form: x squared + y squared + Dx + Ey + F = 0, where D = -2h, E = -2k, and F = h squared + k squared - r squared. A third representation, the parametric form, expresses x and y separately in terms of an angle theta: x = r cos(theta) + h and y = r sin(theta) + k, where theta runs from 0 to 2pi.
How to use this calculator
Select the input mode that matches the information you have. In Center and radius mode, enter the x- and y-coordinates of the center and the radius. In General form mode, enter the three coefficients D, E, and F from the equation x squared + y squared + Dx + Ey + F = 0; the calculator completes the square to find the center and radius. In Three points mode, enter any three points that lie on the circle; the calculator sets up and solves a 2x2 linear system derived from subtracting pairs of point equations to find the center, then computes r as the distance from the center to the first point. All modes output the standard form, general form, parametric equations, center, radius, diameter, area, circumference, x-intercepts, y-intercepts, domain, and range.
Converting between forms: completing the square
To convert from general form to standard form, group the x terms and y terms, then complete the square for each variable. For x squared + Dx, add (D/2) squared to both sides to form (x + D/2) squared. Do the same for y squared + Ey, adding (E/2) squared. This reveals the center as h = -D/2 and k = -E/2, and the radius as r = sqrt(h squared + k squared - F). The three-points method exploits a shortcut: subtracting the circle equation of point 1 from that of point 2 eliminates r squared, leaving a linear equation in h and k. Two such subtractions give a 2x2 system that is solved by Cramer's rule to pinpoint the center.
Area, circumference, intercepts, domain and range
The area enclosed by a circle of radius r is A = pi times r squared. The circumference (perimeter) is C = 2 times pi times r. The x-intercepts are the points where y = 0, found by solving (x - h) squared = r squared - k squared; they exist only when r is at least as large as the vertical distance from the center to the x-axis (that is, |k| is less than or equal to r). Similarly the y-intercepts require |h| to be at most r. The domain of the circle (the set of valid x values) is the closed interval from h - r to h + r, and the range is from k - r to k + r.
Circle equation forms
| Form | Equation | Notes |
|---|---|---|
| Standard | (x - h)² + (y - k)² = r² | Most common; center and radius are explicit |
| General | x² + y² + Dx + Ey + F = 0 | D = -2h, E = -2k, F = h² + k² - r² |
| Parametric | x = r cos(θ) + h, y = r sin(θ) + k | θ ranges 0 to 2π; useful in calculus and physics |
| Origin-centered | x² + y² = r² | Special case when h = 0 and k = 0 |
The three standard ways to write the equation of a circle with center (h, k) and radius r.
Frequently asked questions
What is the standard form of the equation of a circle?
The standard form is (x - h) squared + (y - k) squared = r squared, where (h, k) is the center and r is the radius. It comes directly from the definition of a circle as all points equidistant from the center, combined with the distance formula.
How do I find the equation of a circle from three points?
Substitute each of the three points into the general form x squared + y squared + Dx + Ey + F = 0 to get three equations. Subtracting pairs of equations eliminates F and leaves a 2x2 linear system in D and E (equivalently, in the center coordinates h and k). Solving that system gives the center, and the radius is then the distance from the center to any of the three points. This calculator does all of that automatically.
How do I convert the general form to standard form?
Rearrange x squared + y squared + Dx + Ey + F = 0 so both squared terms are on the left. Group the x terms together and complete the square: add (D/2) squared to both sides. Do the same for the y terms with (E/2) squared. The left side becomes (x + D/2) squared + (y + E/2) squared, and the right side is (D/2) squared + (E/2) squared - F = r squared. The center is at (-D/2, -E/2).
Can a circle have no x-intercepts?
Yes. x-intercepts exist only when the circle crosses or touches the x-axis, which requires the absolute value of the center's y-coordinate k to be less than or equal to the radius. If the center is more than r units away from the x-axis, the circle floats above or below it and has no x-intercepts. The same logic applies to y-intercepts with respect to the y-axis.
What is the parametric form of a circle equation?
The parametric form expresses the x and y coordinates of every point on the circle separately in terms of a single angle theta: x = r cos(theta) + h and y = r sin(theta) + k. As theta sweeps from 0 to 2pi, the point (x, y) traces the full circle exactly once. The parametric form is especially useful in calculus, physics, and computer graphics where motion along the circle must be described.
What is the equation of a circle centered at the origin?
When the center is at the origin, h = 0 and k = 0, so the standard form simplifies to x squared + y squared = r squared. This is the most commonly encountered special case in introductory algebra and trigonometry.