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Circumcenter Calculator

Find the circumcenter of a triangle from its three vertex coordinates. Enter the six coordinate values and read the circumcenter x and y with full working shown.

Enter the three triangle vertices:

Result

(2, 1.5)

The circumcenter is at (2, 1.5) with circumradius 2.5

Quick Answer

A circumcenter calculator finds the point equidistant from all three vertices of a triangle: the centre of the circumscribed circle. For vertices A(0,0), B(4,0), C(0,3), the circumcenter is at (2, 1.5): midpoint of the hypotenuse. The calculator uses the perpendicular bisector intersection formula derived from the three-vertex coordinate inputs.

The Circumcenter: What It Is and How the Formula Works

The circumcenter is the point equidistant from all three vertices of a triangle. It is the centre of the circumscribed circle (circumcircle), the unique circle that passes through all three vertices. The circumcenter is found by intersecting any two of the three perpendicular bisectors of the triangle's sides.

The Circumcenter Formula From Three Vertex Coordinates

D = 2[x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)] Ox = Ux / D, Oy = Uy / D

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

How To Use the Circumcenter Calculator Step By Step

Inputs

  • x₁, y₁: coordinates of the first vertex
  • x₂, y₂: coordinates of the second vertex
  • x₃, y₃: coordinates of the third vertex

Steps

  1. Enter the three vertex coordinates.
  2. The calculator computes D, Ux, and Uy.
  3. If D = 0, the calculator reports that the points are collinear.
  4. Otherwise, it outputs the circumcenter (Ox, Oy) and the circumradius R.
  5. Verify by confirming the distances from O to each vertex are equal.

Circumcenter Worked Example: Three Vertex Coordinates

Finding the circumcenter for vertices P₁(1, 1), P₂(5, 1), P₃(3, 5).

  1. Compute D = 2[1(1−5) + 5(5−1) + 3(1−1)] = 2[−4 + 20 + 0] = 32.
  2. Compute Ux = 2(−4) + 26(4) + 34(0) = 96.
  3. Compute Uy = 2(−2) + 26(−2) + 34(4) = 80.
  4. Circumcenter: Ox = 96/32 = 3.0, Oy = 80/32 = 2.5 → (3.0, 2.5).

Circumcenter = (3.0, 2.5). Circumradius R = 2.5.

When the Circumcenter Calculation Is Most Relevant

The circumcenter is used in geometry problems requiring the circumscribed circle of a triangle, in computational geometry for Delaunay triangulation, in surveying, and in antenna placement problems.

Assumptions

  • The three input points are not collinear (D ≠ 0).
  • All coordinates are real numbers.
  • The formula produces the unique circumcenter.

Limitations

  • Does not compute the incenter, centroid, or orthocenter.
  • Does not verify that the input forms a valid non-degenerate triangle beyond checking D ≠ 0.
  • Does not compute the circumcircle equation beyond reporting centre and radius.

In Practice

A D = 0 result almost always means a data entry error rather than truly collinear points: two identical vertices, or a typo that places two points on the same horizontal or vertical line with the third. Always check your coordinates against the original problem statement if the calculator reports collinear points.

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Frequently Asked Questions: Circumcenter Calculator

What is the circumcenter of a triangle?

The circumcenter is the point equidistant from all three vertices of a triangle. It is the centre of the circumscribed circle: the circle that passes exactly through all three vertices.

Where is the circumcenter relative to the triangle?

For an acute triangle, the circumcenter is inside the triangle. For a right triangle, it lies at the midpoint of the hypotenuse. For an obtuse triangle, it lies outside the triangle.

How is the circumcenter different from the centroid?

The centroid is the arithmetic average of the three vertex coordinates: ((x1+x2+x3)/3, (y1+y2+y3)/3). It is the centre of mass of the triangle. The circumcenter is the centre of the circumscribed circle.

What does D = 0 mean in the formula?

D = 0 means the three points are collinear: they lie on the same straight line and do not form a triangle. No circumscribed circle exists for three collinear points.

What is the circumradius?

The circumradius R is the radius of the circumscribed circle, equal to the distance from the circumcenter O to any of the three vertices: R = √((Ox − x₁)² + (Oy − y₁)²).

Sources

Last updated: . Reviewed for accuracy against the formula shown above.