Is the circumcenter inside, on or outside the triangle?

Inside an acute triangle, exactly on the midpoint of the hypotenuse in a right triangle, and outside beyond the longest side in an obtuse one.

Why is the right triangle case so simple?

Thales: the hypotenuse subtends a right angle, so it is a diameter and its midpoint is the center. A(0, 0), B(4, 0), C(0, 3) gives (2, 1.5) with R = 2.5 without solving anything.

How can I cross-check the radius against the sides?

With R = abc ÷ 4K: sides 3, 4, 5 and area 6 give 60 / 24 = 2.5, the same as the coordinates.

📍 Circumcenter Calculator

Calculate the circumcenter (center of circumscribed circle) coordinates of a triangle.

Vertex A:x₁ y₁

Vertex B:x₂ y₂

Vertex C:x₃ y₃

What is the Circumcenter?

The point where the three perpendicular bisectors of a triangle meet. All three vertices are equidistant from the circumcenter, and this distance is the circumscribed circle radius (R).

Circumcenter Coordinate Formula

D = 2[x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)]

Ox = [(x₁²+y₁²)(y₂−y₃) + (x₂²+y₂²)(y₃−y₁) + (x₃²+y₃²)(y₁−y₂)] / D

Oy = [(x₁²+y₁²)(x₃−x₂) + (x₂²+y₂²)(x₁−x₃) + (x₃²+y₃²)(x₂−x₁)] / D

Position of the Circumcenter

Acute triangle: inside the triangle

Right triangle: midpoint of the hypotenuse

Obtuse triangle: outside the triangle

Circumscribed Circle Radius

R = |OA| = |OB| = |OC|

R = abc / (4S)

Worked examples

Vertices A(0, 0), B(4, 0) and C(0, 3) place the circumcenter at (2, 1.5), the midpoint of the hypotenuse, with R = 2.5 and a circle area of π × 6.25 ≈ 19.635.

An isosceles triangle with A(0, 0), B(6, 0) and C(3, 4) has its circumcenter at (3, 0.875), a radius of 3.125 and a circle area of about 30.6796.

What the coordinates assume

The three points must not lie on one line: a collinear set has no finite circumcenter, so the calculator reports an error instead. Coordinates are read as Cartesian values in a single unit, giving the radius that unit and the area its square. Because the result comes from solving the perpendicular-bisector equations, extremely large coordinates lose precision in floating point.

Frequently asked questions

Is the circumcenter inside, on or outside the triangle?: Inside an acute triangle, exactly on the midpoint of the hypotenuse in a right triangle, and outside beyond the longest side in an obtuse one.
Why is the right triangle case so simple?: Thales: the hypotenuse subtends a right angle, so it is a diameter and its midpoint is the center. A(0, 0), B(4, 0), C(0, 3) gives (2, 1.5) with R = 2.5 without solving anything.
How can I cross-check the radius against the sides?: With R = abc ÷ 4K: sides 3, 4, 5 and area 6 give 60 / 24 = 2.5, the same as the coordinates.