Center of a Circle Calculator
Find the center coordinates (h, k) and radius of any circle from three different starting points: the standard-form equation, the general (expanded) equation, or the coordinates of three points that lie on the circle. Choose a method below, enter your values, and the calculator shows the full worked solution instantly.
Formula
Worked example
General form x² + y² - 6x + 4y - 12 = 0: D = -6, E = 4, F = -12 so h = 3, k = -2 and r² = 9 + 4 + 12 = 25, giving center (3, -2) and r = 5.
What is the center of a circle?
The center of a circle is the single fixed point that is equidistant from every point on the circle's edge. That common distance is the radius. In coordinate geometry, if the center is at the point (h, k) and the radius is r, the circle is described by the standard equation (x - h)² + (y - k)² = r². Knowing the center and radius completely defines the circle and lets you convert between all common equation forms, check whether a point lies inside or outside the circle, and find tangent lines or chord lengths.
Three ways to find the center
This calculator supports all three common starting points. If you already have the standard form (x - h)² + (y - k)² = r², the center (h, k) is read off directly and the radius is the square root of the right-hand side. If you have the general (expanded) form x² + y² + Dx + Ey + F = 0, complete the square in x and y: the center is at h = -D/2, k = -E/2, and r² = h² + k² - F. If you have three distinct, non-collinear points that lie on the circle, the center is the circumcenter of the triangle those points form - the unique point equidistant from all three, found by intersecting the perpendicular bisectors of any two chords.
Converting between standard form and general form
Starting from standard form (x - h)² + (y - k)² = r², expand the squares: x² - 2hx + h² + y² - 2ky + k² = r². Rearranging to equal zero gives x² + y² + (-2h)x + (-2k)y + (h² + k² - r²) = 0. Reading off the coefficients: D = -2h, E = -2k, and F = h² + k² - r². To go the other way, given D, E and F, recover h = -D/2, k = -E/2 and r² = h² + k² - F.
The three-point (circumcenter) method
Any three non-collinear points in a plane determine a unique circle. The center is the circumcenter of the triangle formed by the three points. To find it, take the midpoints of any two chords (line segments connecting pairs of the three points), then draw perpendicular lines through each midpoint. The perpendicular bisectors of two chords always intersect at the circle's center. This calculator handles the algebra for you by solving the 2x2 linear system that arises. If you enter three collinear points (which cannot all lie on a single circle), no valid answer exists and the calculator returns no result.
Circle equation forms compared
| Form | Equation | Center from equation | Radius from equation |
|---|---|---|---|
| Standard | (x - h)² + (y - k)² = r² | (h, k) - read off directly | r = sqrt(right-hand side) |
| General | x² + y² + Dx + Ey + F = 0 | h = -D/2, k = -E/2 | r = sqrt(h² + k² - F) |
| Three points | Three (x, y) pairs on circle | Circumcenter of triangle | Distance center to any point |
| Diameter endpoints | (x - x1)(x - x2) + (y - y1)(y - y2) = 0 | Midpoint of diameter | Half the distance x1 to x2 |
All three forms describe the same circle. Use whichever matches what your problem gives you.
Frequently asked questions
How do I find the center of a circle from its equation?
It depends on which form of the equation you have. In standard form (x - h)² + (y - k)² = r², the center (h, k) is written directly into the equation. In general form x² + y² + Dx + Ey + F = 0, the center is at h = -D/2 and k = -E/2. Enter your coefficients into this calculator and it will find the center and radius for you.
How do I find the center of a circle given three points?
The center of a circle passing through three points is the circumcenter of the triangle formed by those points. It is the intersection of the perpendicular bisectors of any two of the three chords. This calculator automates that algebra: enter the three (x, y) coordinates, select the three-point method, and it returns the center and radius.
What is the difference between standard form and general form of a circle?
Standard form (x - h)² + (y - k)² = r² shows the center (h, k) and radius r explicitly. General form x² + y² + Dx + Ey + F = 0 is the expanded and rearranged version - it looks less structured but is the form that often arises from algebraic manipulation. Both forms describe exactly the same circle. You can convert between them using the relationships D = -2h, E = -2k, and F = h² + k² - r².
What happens if I enter three collinear points?
Three collinear points (all on the same line) cannot all lie on a circle - no finite-radius circle passes through them. In that case the perpendicular bisectors of the chords are parallel and never intersect, so the system has no solution. The calculator will return no result, which is the correct mathematical answer.
How is the radius related to the standard-form equation?
In the standard form (x - h)² + (y - k)² = r², the number on the right-hand side is r² (radius squared), not the radius itself. The radius is the square root of that number. For example, if the equation is (x - 3)² + (y + 2)² = 25, then r² = 25 and r = 5.