Circumcenter Calculator
Enter the x and y coordinates of a triangle's three vertices (decimals and negatives welcome) and get the circumcenter, circumradius, and the full circumcircle equation, rounded to 3 decimals.
Example: with Vertex A — x₁ 0 · Vertex A — y₁ 0 · Vertex B — x₂ 6 · Vertex B — y₂ 0 · Vertex C — x₃ 0 → Circumcenter (x, y): (3, 4).
- Circumradius R5 units
- Circumcircle equation(x - 3)² + (y - 4)² = 25
- Where it sitsRight triangle — the circumcenter is the midpoint of the hypotenuse.
Computed by the calculator below using its default values. Change any input to see your own numbers.
The circumcenter is the meeting point of the three perpendicular bisectors — the only point equidistant from all three vertices.
What the circumcenter is
Every side of a triangle has a perpendicular bisector — the line of points equally far from that side's two endpoints. All three bisectors cross at one spot: the circumcenter, the only point equidistant from all three vertices. That shared distance is the circumradius R, and the circle it draws — the circumcircle — passes exactly through A, B, and C.
Algebraically, setting distance-to-A equal to distance-to-B and distance-to-C gives two linear equations; solving them produces the determinant formula this page uses. No trigonometry is needed, and the answer is exact before display rounding.
Reading the result
Where the circumcenter lands tells you the triangle's shape at a glance. Acute triangles keep it inside; obtuse triangles push it out beyond the longest side; and in a right triangle it lands exactly on the midpoint of the hypotenuse — which is why the default example (a 6-8-10 right triangle) returns (3, 4) with R = 5. If you get a huge radius from a skinny, nearly-flat triangle, that is real: near-collinear points genuinely need an enormous circle.
How it’s calculated
D = 2[x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)]. Center: 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 — the intersection of two perpendicular bisectors. R is the distance from the center to any vertex. Math runs at full precision; display rounds to 3 decimals.
Plane (2D) coordinates only; if D is effectively zero the points are collinear and no circumcircle exists.
Where the circumcenter lands
| Triangle type | Circumcenter location |
|---|---|
| Acute | Inside the triangle |
| Right | Exactly at the midpoint of the hypotenuse |
| Obtuse | Outside, beyond the longest side |
| Equilateral | Coincides with the centroid, incenter, and orthocenter |
Classical Euclidean geometry — the circumcenter is the intersection of the perpendicular bisectors of the sides.
Common mistakes
- Expecting the circumcenter inside every triangle — obtuse triangles put it outside.
- Mixing up the circumcenter (equidistant from the vertices) with the incenter (equidistant from the sides).
- Entering two identical points or three collinear points — no unique circle exists through them.
- Rounding intermediate values when working by hand; small errors blow up when the triangle is nearly flat.
Frequently asked questions
What is the circumcenter formula?
With vertices (x₁,y₁), (x₂,y₂), (x₃,y₃), compute D = 2[x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)]. Then Ux = [(x₁² + y₁²)(y₂ − y₃) + (x₂² + y₂²)(y₃ − y₁) + (x₃² + y₃²)(y₁ − y₂)]/D, and Uy is the mirror expression using x-differences. It comes from intersecting two perpendicular bisectors.
Why is my circumcenter outside the triangle?
The triangle is obtuse. The circumcircle still passes through all three vertices, but its center sits beyond the longest side. Only acute triangles keep the center inside.
What is the circumradius?
The distance from the circumcenter to any vertex — all three are equal by construction. It also equals abc/(4K), where a, b, c are the side lengths and K is the triangle's area.
How is the circumcenter different from the centroid?
The centroid averages the three vertices and is the balance point; the circumcenter is equidistant from the vertices. They only coincide for an equilateral triangle, and both lie on the Euler line along with the orthocenter.