Circumcircle Calculator
What the Circumcircle Calculator Finds
A circumcircle is the circle that passes through all three vertices of a triangle. For any three points that are not on the same straight line, there is exactly one such circle. This calculator takes the coordinates of points A, B, and C and returns the circumcenter, the radius, and the standard-form equation of the circle.
The result is useful in coordinate geometry, CAD sketches, robotics paths, surveying problems, game physics, and triangle analysis. If the three points are nearly collinear, the circle becomes extremely large and numerically unstable, so the calculator reports that the circumcircle is undefined instead of returning a misleading result.
How the Inputs Work
Enter each point as an ordered pair. The x and y coordinates can be in any consistent linear unit: meters, feet, pixels, grid units, or plain coordinate units. The calculator does not convert units; it assumes all six inputs use the same coordinate system. If your drawing is in meters, the radius is in meters. If the input is in pixels, the radius is in pixels.
The order of the three points does not change the circle. You can enter A, B, and C clockwise, counterclockwise, or in any other order. What matters is that the points are distinct and not collinear.
Formula Used
For points A = (x1, y1), B = (x2, y2), and C = (x3, y3), the calculator first computes:
If D is zero, the points are collinear and no finite circumcircle exists. Otherwise, the center coordinates (h, k) are:
After the center is known, the radius is the distance from the center to any one of the three points:
Worked Example
Suppose the three points are A = (0, 0), B = (4, 0), and C = (0, 3). These points form a right triangle. The midpoint of the hypotenuse is (2, 1.5), and for a right triangle that midpoint is also the circumcenter. The radius is the distance from (2, 1.5) to any vertex:
r = sqrt((2 - 0)2 + (1.5 - 0)2) = 2.5
The circle equation is therefore:
(x - 2)2 + (y - 1.5)2 = 6.25
Interpreting Results
The center tells you where the circle is located, while the radius tells you how far the circle extends from that center. The equation is shown in standard form:
(x - h)2 + (y - k)2 = r2
If the center lies outside the triangle, the triangle is obtuse. If the center lies inside the triangle, the triangle is acute. If the center lies exactly on one side, the triangle is right. This makes the circumcenter useful not only for drawing the circle, but also for understanding the geometry of the triangle itself.
Scenario Comparison
| Points | Geometry | Circumcenter behavior |
|---|---|---|
| (0,0), (4,0), (0,3) | Right triangle | Center is the midpoint of the hypotenuse. |
| (0,0), (3,0), (1,3) | Acute triangle | Center falls inside the triangle. |
| (0,0), (4,0), (1,1) | Obtuse triangle | Center falls outside the triangle. |
| (0,0), (1,1), (2,2) | Collinear points | No finite circumcircle exists. |
Limitations
Coordinate values are rounded in the displayed result, so very small differences may disappear visually even though the calculation uses full JavaScript precision internally. Points that are extremely close together or nearly collinear can produce a very large radius, because the perpendicular bisectors intersect far away from the triangle. In those cases, small measurement errors can move the center dramatically.
Use consistent units and avoid entering duplicate points. If you are using measured field data, repeat the calculation with upper and lower measurement tolerances to see how sensitive the circumcenter is to small coordinate changes.