Circumcenter Calculator

Enter the (x, y) coordinates of a triangle's three vertices to find its circumcenter, circumradius, and whether the triangle is acute, right, or obtuse.

Quick Facts

Definition
Intersection of the three perpendicular bisectors
Equidistant from all three vertices — the center of the circumscribed circle.
Location rule
Inside = acute, on the hypotenuse = right, outside = obtuse
The circumcenter's position reveals the triangle's angle type.

Your Results

Calculated
Circumcenter Ux
-
X-coordinate of the circumcenter
Circumcenter Uy
-
Y-coordinate of the circumcenter
Circumradius R
-
Distance from circumcenter to each vertex
Triangle type
-
Based on circumcenter position

Ready

Enter three vertex coordinates, then press Calculate.

Formula and method for the circumcenter

The circumcenter of a triangle is the point that is equidistant from all three vertices. It is the unique intersection of the three perpendicular bisectors of the triangle's sides, and it marks the center of the circumcircle — the circle that passes through all three vertices.

How the calculation works

Given vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), the circumcenter (Ux, Uy) is found by solving the system formed by two perpendicular bisectors:

  • D = 2 × [x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)]
  • Ux = [(x₁²+y₁²)(y₂−y₃) + (x₂²+y₂²)(y₃−y₁) + (x₃²+y₃²)(y₁−y₂)] / D
  • Uy = [(x₁²+y₁²)(x₃−x₂) + (x₂²+y₂²)(x₁−x₃) + (x₃²+y₃²)(x₂−x₁)] / D

The circumradius R is simply the distance from (Ux, Uy) to any one of the three vertices — the result is identical for all three by construction. If D = 0, the three points are collinear and no triangle (or circumcenter) exists.

Reading the triangle type from the circumcenter

The circumcenter's position relative to the triangle tells you its angle classification: it falls inside an acute triangle (all angles under 90°), lands exactly on the midpoint of the hypotenuse for a right triangle, and sits outside an obtuse triangle (one angle over 90°). This calculator determines the type directly from the side lengths using the law of cosines relationship (a² + b² vs. c² for the longest side c), which avoids floating-point ambiguity near the boundary.

Common mistakes

  • Repeating or collinear points: if two vertices coincide, or all three lie on one line, there is no valid triangle and D = 0.
  • Confusing circumcenter with centroid or incenter: the centroid (average of the three vertices) and incenter (intersection of angle bisectors) are different triangle centers and generally sit at different points than the circumcenter, except in equilateral triangles where all three coincide.
  • Mixed units or scales: use the same unit for every x and y coordinate; the calculator treats all inputs as plain numbers in one consistent coordinate system.

Real-world applications

  • Finding the center and radius of a circle that passes through three known points (surveying, CAD, and CNC work)
  • Placing a facility or sensor equidistant from three fixed locations
  • Triangle classification in computational geometry and mesh-quality checks
  • Geometry coursework on perpendicular bisectors, triangle centers, and circumscribed circles

Frequently Asked Questions

What is the circumcenter of a triangle?
The circumcenter is the point equidistant from all three vertices of a triangle. It is found where the perpendicular bisectors of the three sides intersect, and it is the center of the circumscribed circle (circumcircle) that passes through all three vertices.
How is the circumcenter calculated?
Given vertices (x1,y1), (x2,y2), (x3,y3), the circumcenter (Ux, Uy) is found by solving the intersection of two perpendicular bisectors: D = 2[x1(y2−y3) + x2(y3−y1) + x3(y1−y2)], Ux = [(x1²+y1²)(y2−y3) + (x2²+y2²)(y3−y1) + (x3²+y3²)(y1−y2)] / D, and Uy = [(x1²+y1²)(x3−x2) + (x2²+y2²)(x1−x3) + (x3²+y3²)(x2−x1)] / D.
Why does the calculator say the points do not form a triangle?
If the three points are collinear (all on one straight line) or two points coincide, the denominator D in the circumcenter formula becomes zero and no finite circumcenter exists. The calculator flags this as an invalid triangle.
How does the circumcenter show whether a triangle is acute, right, or obtuse?
The circumcenter's location relative to the triangle reveals its angle type: it lies inside an acute triangle, exactly on the hypotenuse (its midpoint) for a right triangle, and outside an obtuse triangle. The calculator determines this directly from the side lengths using the Pythagorean relationship.

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