About

The circumcenter of a triangle is the unique point equidistant from all three vertices. It lies at the intersection of the three perpendicular bisectors. For acute triangles the circumcenter falls inside the triangle. For right triangles it sits on the hypotenuse midpoint. For obtuse triangles it lies outside. Miscalculating this point leads to incorrect circumscribed circle fits, which propagates errors in computational geometry pipelines such as Delaunay triangulation, mesh generation, and navigation coordinate systems. This tool computes the exact circumcenter coordinates (O_x, O_y) and the circumradius R using the perpendicular bisector intersection method, and validates against degenerate (collinear) inputs where no circumcircle exists.

The computation assumes a standard Euclidean plane with Cartesian coordinates. Numerical precision is limited to IEEE 754 double-precision floating point. For nearly-collinear triangles (area approaching 0), results may exhibit significant rounding artifacts. The tool does not handle spherical or hyperbolic geometry.

Formulas

The circumcenter O is found by solving the intersection of two perpendicular bisectors. Given vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃), the circumcenter coordinates are computed via:

O_x = 12D ⋅ [ (x₁² + y₁²)(y₂ − y₃) + (x₂² + y₂²)(y₃ − y₁) + (x₃² + y₃²)(y₁ − y₂) ]

O_y = 12D ⋅ [ (x₁² + y₁²)(x₃ − x₂) + (x₂² + y₂²)(x₁ − x₃) + (x₃² + y₃²)(x₂ − x₁) ]

where D is the determinant:

D = x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)

When D = 0, the points are collinear and no circumcircle exists. The circumradius is computed as:

R = a ⋅ b ⋅ c4 ⋅ S

where a, b, c are the side lengths and S is the triangle area computed by the shoelace formula: S = |D|2.

Reference Data

Triangle Type	Circumcenter Location	Angle Condition	Example Vertices	Circumcenter	Circumradius
Equilateral	Centroid (inside)	All angles = 60°	(0,0), (4,0), (2,3.464)	(2, 1.155)	2.309
Right (at C)	Hypotenuse midpoint	One angle = 90°	(0,0), (4,0), (0,3)	(2, 1.5)	2.5
Acute Isosceles	Inside, on axis of symmetry	All angles < 90°	(0,0), (6,0), (3,5)	(3, 1.7)	3.481
Obtuse Scalene	Outside (opposite obtuse vertex)	One angle > 90°	(0,0), (5,0), (1,1)	(2.5, −3.5)	4.301
Acute Scalene	Inside	All angles < 90°	(1,1), (5,2), (3,6)	(3.35, 3.15)	2.693
Degenerate (collinear)	Undefined	Area = 0	(0,0), (1,1), (2,2)	N/A	∞
Very Thin Obtuse	Far outside	Largest angle → 180°	(0,0), (10,0), (5,0.1)	(5, 124.975)	124.98
Right Isosceles	Hypotenuse midpoint	One angle = 90°, two = 45°	(0,0), (4,0), (0,4)	(2, 2)	2.828
30-60-90 Triangle	Inside, closer to hypotenuse	30°, 60°, 90°	(0,0), (4,0), (0,2.309)	(2, 1.155)	2.309
Large Coordinates	Inside	Arbitrary acute	(100,200), (400,150), (250,500)	(261.07, 272.02)	174.56
Negative Coordinates	Inside	Arbitrary acute	(−3,−2), (4,−1), (1,5)	(1.045, 0.955)	4.261

Frequently Asked Questions

When the signed area of the triangle equals zero (the determinant D = 0), the three points lie on a single line. No circumscribed circle exists because you cannot pass a unique circle through three collinear points. The circumradius approaches infinity. The calculator detects this condition and reports the input as degenerate.

The circumcenter is the intersection of perpendicular bisectors. When one interior angle exceeds 90°, the perpendicular bisectors of the two sides forming that angle intersect on the exterior side opposite the obtuse vertex. This is a geometric necessity: the circumscribed circle must still pass through all three vertices, and its center shifts outward to maintain equal radial distance.

When the triangle area is extremely small (nearly collinear points), the determinant D approaches 0. Division by a near-zero value amplifies rounding errors in IEEE 754 double-precision arithmetic. The calculator uses an epsilon threshold of 1e−10 to classify inputs as degenerate. For triangles with area below this threshold, results are unreliable and the tool warns accordingly.

No. The circumcenter (intersection of perpendicular bisectors) is equidistant from all vertices. The incenter (intersection of angle bisectors) is equidistant from all sides. The centroid (intersection of medians) is the center of mass. These three points coincide only in equilateral triangles. For all other triangles they are distinct points.

Yes. The formulas operate on arbitrary real-valued Cartesian coordinates. Negative values are valid. Very large values (beyond ±1e12) may reduce precision due to floating-point representation limits, but standard engineering-scale coordinates (up to millions) produce accurate results to at least 6 significant digits.

In Delaunay triangulation, the circumscribed circle of every triangle in the mesh must contain no other vertices from the point set. The circumcenter and circumradius define this empty-circle condition. Errors in circumcenter computation directly propagate into incorrect triangulations, affecting applications in finite element analysis, terrain modeling, and mesh generation.