Three-Point Circle Finder
Enter three coordinates that lie on a curve — from recovered monuments, for instance — to solve for the circle’s center, radius, and the arcs and chords between the points.
Field aid, not a certified result. Points that are nearly collinear produce an unstable, unreliable radius even before the calculator flags true collinearity — use well-spread points where possible.
Three points on the curve
Check a fourth point against this circle (optional)
Enter all three points to solve for the circle.
| Pair | Chord | Center angle | Minor arc length |
|---|
Center angle and arc length use the direct angle between each pair of points as seen from the center — always the shorter (0°–180°) way around, regardless of point order.
Solving a curve from three known points
Quick answer: Any three points that aren’t in a straight line define exactly one circle. Given their coordinates, the circle’s center is the point equidistant from all three — found where the perpendicular bisectors of the segments between them intersect — and the radius is simply the distance from that center to any of the points.
This comes up often in resurveying: a curve’s original center point and radius were never monumented, or the record has been lost, but two or three points along the curve itself — recovered monuments, found pins — still exist. Rather than guessing the curve’s parameters, the three-point solution recovers the exact circle those points lie on, which can then be compared against record bearings and distances to check for consistency.
The direct angle between any two points as seen from the center — found from the vectors to each point, not from compass bearings — avoids the wraparound ambiguity that azimuth subtraction can introduce, and always gives the correct angle between 0° and 180°. Multiplying that angle (in radians) by the radius gives the arc length along the shorter path between the two points.
Why nearly-collinear points are a problem
As three points approach a straight line, the circle that passes through them grows enormously — in the limit, an infinite radius. Points that are only slightly off a straight line can still produce a huge, unstable radius that amplifies small measurement errors. Spreading the three points out, especially the middle one away from the chord of the outer two, gives a far more reliable result.
Frequently asked questions
Why do I need exactly three points?
Three non-collinear points are the minimum needed to uniquely define a circle. Two points alone could lie on infinitely many different circles.
What does it mean if the points are “collinear”?
It means all three fall on (or extremely close to) a single straight line, so no finite circle passes through them. The calculator will flag this rather than returning an unstable or meaningless radius.
Does the order I enter the points in matter?
Not for the center or radius — those depend only on the three positions. Order only affects the reported clockwise/counterclockwise direction, which describes the rotational sense of P1 → P2 → P3.
What is the fourth-point check for?
It’s a way to verify a fourth recovered point against the curve solved from the first three — useful for confirming that all four monuments genuinely lie on the same circular curve.
Why use the center angle instead of bearings to get the arc length?
Subtracting compass bearings directly can give the wrong angle when it wraps past 0°/360°. Computing the angle between the two position vectors from the center avoids that ambiguity entirely and always gives the correct value.
Does this replace a licensed survey?
No. This is a calculation aid for checking and reconstructing curve geometry. Any result used for a legal boundary or recorded plat should be verified and certified by a licensed surveyor.
