Circle Center Calculator — Find Center & Radius from 3 Points
Enter any three points that lie on a circle to find its center (h, k) and radius — calculated by solving the perpendicular-bisector system.
Distance from center (h, k) to each of the three input points
- 1
Linear system determinant
8 × 6 − 0 × 0 = 48Non-zero determinant confirms the three points are not collinear. - 2
Center x (h)
(16 × 6 − 9 × 0) ÷ 48 = 2 - 3
Center y (k)
(8 × 9 − 0 × 16) ÷ 48 = 1.5 - 4
Radius = distance from center to P₁
√((0 − 2)² + (0 − 1.5)²) = 2.5000
How does this calculator work?
Three non-collinear points uniquely fix a circle. Cancel the squared terms in the equal-distance equations to get a 2×2 linear system; solve for center (h, k) by Cramer's rule. Radius = distance from center to any input point. Result is displayed as the standard-form equation and plotted.
Formula
How this is calculated
Any three non-collinear points lie on exactly one circle. Its center is equidistant from all three — which is the same as the circumcenter of the triangle they form. Setting the squared distance from an unknown center (h, k) to two pairs of points equal gives a linear equation in h and k each time (the squared terms cancel). Two such equations form a 2×2 system solvable by Cramer's rule.
The calculator computes coefficients a₁, b₁, c₁ from points 1 and 2, and a₂, b₂, c₂ from points 1 and 3, evaluates the determinant, then solves for h and k. The radius is the Euclidean distance from the center to any of the three input points. The result is algebraically exact, aside from floating-point rounding.
If the three points are collinear (all on one line) the determinant is zero — no unique circle exists. Move at least one point off the line to get a valid result.
Frequently asked questions
The circumcenter is the point equidistant from all three vertices of a triangle — it is the center of the circle that passes through (circumscribes) all three vertices. For acute triangles it lies inside; for obtuse triangles it lies outside.
Two points leave one degree of freedom: infinitely many circles of different radii can pass through them. A third point removes that freedom, uniquely fixing both center and radius — provided the points are not collinear.
The points you entered are collinear (all on a straight line). A straight line is a circle of infinite radius. Adjust at least one point so they form a triangle with non-zero area.
Also known as
TG we-Calculate Editorial Team. (2026). Circle Center Calculator — Find Center & Radius from 3 Points [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/circle-center-calculator
TG we-Calculate Editorial Team. "Circle Center Calculator — Find Center & Radius from 3 Points." TG we-Calculate. 2026. https://we-calculate.com/calculator/circle-center-calculator.
TG we-Calculate Editorial Team, "Circle Center Calculator — Find Center & Radius from 3 Points," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/circle-center-calculator
@misc{wecalculate_circle_center_calculator, title = {Circle Center Calculator — Find Center & Radius from 3 Points}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/circle-center-calculator}}, year = {2026}, note = {TG we-Calculate} }
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