Circumcenter of a Triangle Calculator — Circumscribed Circle
Enter the coordinates of three triangle vertices to find the circumcenter (the point equidistant from all three) and circumradius — the radius of the unique circle that passes through all three vertices.
Radius of the circle passing through all three vertices
- 1
Denominator D
2 × (0(0−5) + 6(5−0) + 2(0−0)) = 60D equals twice the signed triangle area; zero means the points are collinear. - 2
Vertex squared norms
a² = 0, b² = 36, c² = 29 - 3
Circumcenter x
(0×(0−5) + 36×(5−0) + 29×(0−0)) ÷ 60 = 3 - 4
Circumcenter y
(0×(2−6) + 36×(0−2) + 29×(6−0)) ÷ 60 = 1.7 - 5
Circumradius R
√((0−3)² + (0−1.7)²) = 3.4482
How does this calculator work?
The circumcenter is where the three perpendicular bisectors of a triangle's sides meet — equidistant from all three vertices. Formula: D = 2[x₁(y₂−y₃)+…], then O_x and O_y from vertex squared-norms divided by D. Circumradius R = |OA|. Enter coordinates to get O and R instantly.
Formula
How this is calculated
The circumcenter of a triangle is the point equidistant from all three vertices — it is the centre of the unique circumscribed circle (circumcircle) that passes through A, B and C. It is found by intersecting any two of the three perpendicular bisectors of the sides; they always meet at a single point for a non-degenerate triangle.
The coordinate formula uses the squared norms of each vertex. Setting D = 2[x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)] (twice the signed area), the circumcenter coordinates are O_x = [(x₁²+y₁²)(y₂−y₃) + (x₂²+y₂²)(y₃−y₁) + (x₃²+y₃²)(y₁−y₂)] / D and similarly O_y. The circumradius R = distance from O to any vertex. If D = 0 the three points are collinear and no circumcircle exists.
For acute triangles the circumcenter lies inside the triangle; for obtuse triangles it lies outside; for right triangles it is exactly the midpoint of the hypotenuse (since the hypotenuse is a diameter of the circumcircle).
Frequently asked questions
For acute triangles it lies inside; for obtuse triangles it lies outside (beyond the longest side); for right triangles it lies exactly at the midpoint of the hypotenuse. This is a direct consequence of the Inscribed Angle Theorem.
No — these are distinct triangle centres. The centroid (intersection of medians) is the centre of mass. The incentre (intersection of angle bisectors) is the centre of the inscribed circle. The circumcenter (intersection of perpendicular bisectors) is the centre of the circumscribed circle. They coincide only for equilateral triangles.
Collinear points lie on a single straight line, so they do not form a triangle and no unique circle passes through all three. The calculator returns no result in this case (D = 0 in the formula).
Also known as
TG we-Calculate Editorial Team. (2026). Circumcenter of a Triangle Calculator — Circumscribed Circle [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/circumcenter-of-a-triangle-calculator
TG we-Calculate Editorial Team. "Circumcenter of a Triangle Calculator — Circumscribed Circle." TG we-Calculate. 2026. https://we-calculate.com/calculator/circumcenter-of-a-triangle-calculator.
TG we-Calculate Editorial Team, "Circumcenter of a Triangle Calculator — Circumscribed Circle," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/circumcenter-of-a-triangle-calculator
@misc{wecalculate_circumcenter_of_a_triangle_calculator, title = {Circumcenter of a Triangle Calculator — Circumscribed Circle}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/circumcenter-of-a-triangle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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