Circle From 3 Points Calculator
This circle from 3 points calculator — also called a circle through 3 points calculator, circumscribed circle calculator, or circumcircle calculator — works out the unique circle that passes through any three coordinates you give it. Enter the X and Y values for three points, tap Calculate, and it returns the circle's standard equation, general equation, center, and radius — plus a plot showing the points and the circle together. Everything runs in your browser: no upload, no account, no ads.
The Formula for a Circle Through 3 Points
In coordinate geometry, any three non-collinear points (x₁,y₁), (x₂,y₂), and (x₃,y₃) define exactly one circle, and the circle passing through all three has its center at the circumcenter of the triangle they form. The center coordinates (h, k) come from a determinant-based formula built from the three points' coordinates and their squared distances from the origin, and the radius r is then just the distance from that center to any one of the three points. This calculator runs that algebra the instant you tap Calculate, so there's no determinant to expand by hand.
Standard Form vs. General Form
Once the center (h, k) and radius r are known, the circle can be written two equivalent
ways. The standard form, (x - h)² + (y - k)² = r², shows the
center and radius directly — useful for graphing or reading off the geometry at a glance.
The general form, x² + y² + Dx + Ey + F = 0, expands that same
equation into a single polynomial (with D = -2h, E = -2k, and F = h² + k² - r²) — the shape
most textbooks and coordinate-geometry problems expect as a final answer. This calculator
shows both forms side by side, with negative coordinates handled correctly so you never see
an awkward double negative like "x - -4".
Worked Example: Circumcircle of a Triangle
Center: (2, 2) · Radius: 2.828
Standard form: (x - 2)² + (y - 2)² = 8
General form: x² + y² - 4x - 4y = 0
These three points form a right triangle with legs of length 4, and its hypotenuse is a diameter of the circumcircle — which is why the radius comes out to half of 4√2, or about 2.828. That relationship (a right angle always sits on the circle whose diameter is the hypotenuse) is a useful sanity check whenever your three points include a right angle, and it's a quick way to check any circumcircle formula by hand before trusting the result.
When Three Points Don't Define a Circle
Three points define exactly one circle only if they are not collinear — if they don't all sit on the same straight line. Collinear points (or two points that are identical) can't form a triangle, so there's no finite circumcenter and no circle passes through all three. This calculator checks for that case automatically: if your three points are collinear or repeated, it tells you plainly instead of returning a meaningless or broken result.
Frequently Asked Questions
How do you find the equation of a circle from 3 points?
Solve for the circumcenter (h, k) and radius r using the determinant method, then substitute them into (x - h)² + (y - k)² = r². Type the three points into this circle equation from 3 points calculator and it does the algebra for you, showing both the standard and general form of the equation instantly.
How do you find the center of a circle from 3 points?
The center is the circumcenter of the triangle formed by the three points — the point equidistant from all three, found where the perpendicular bisectors of the triangle's sides meet. Use this tool to find the circle center from 3 points directly from the coordinates; it shows the center as (h, k).
How do you find the radius of a circle from 3 points?
Once you have the center (h, k), the radius is just the distance from the center to any one of the three points: r = √((x₁ - h)² + (y₁ - k)²) — the standard radius of a circle passing through 3 points formula. This calculator computes the center first, then the radius, automatically.
What is the circumcircle of a triangle?
The circumcircle is the unique circle that passes through all three vertices of a triangle. Every triangle has exactly one — its center (the circumcenter) is equidistant from all three corners, which is exactly what a circle through 3 points calculates.
Can any three points form a circle?
Any three points that are NOT all on the same straight line define exactly one circle. If the three points are collinear (or two of them are identical), there is no finite circle that passes through all three — this calculator detects that case and tells you instead of guessing.
