Circumcenter of a Triangle Calculator

Find the circumcenter from 3 vertex coordinates using the perpendicular bisector intersection method. Also computes the circumradius.

Enter Triangle Vertices

Vertex A
Vertex B
Vertex C

Result

Circumcenter
(3, 1.75)
coordinates
Circumcenter X3
Circumcenter Y1.75
Circumradius (R)3.25
Side a (BC)5
Side b (AC)5
Side c (AB)6
Triangle Area12

Step-by-Step Solution

Circumcenter = intersection of perpendicular bisectors

What is the Circumcenter of a Triangle?

The circumcenter of a triangle is the point where the perpendicular bisectors of all three sides intersect. It is equidistant from all three vertices of the triangle, making it the center of the circumscribed circle (circumcircle) -- the unique circle that passes through all three vertices. The distance from the circumcenter to any vertex is called the circumradius (R).

Circumcenter Properties and Formulas

Circumradius Formula

Using the sides and area of the triangle.

R = abc / (4K)

Perpendicular Bisector Method

Find midpoints and slopes, then solve the system of equations.

Solve 2 bisector equations

Circumcenter for Right Triangle

The circumcenter lies at the midpoint of the hypotenuse.

O = midpoint of hypotenuse

Acute Triangle

Circumcenter lies inside the triangle.

All angles < 90°

Obtuse Triangle

Circumcenter lies outside the triangle.

One angle > 90°

Equilateral Triangle

Circumcenter coincides with centroid and incenter.

R = a / √3

How to Find the Circumcenter

Method 1: Perpendicular Bisector Intersection

The most common method to find the circumcenter involves these steps: (1) Find the midpoints of at least two sides of the triangle. (2) Determine the slopes of those sides, then compute the negative reciprocal (perpendicular slope). (3) Write the equation of each perpendicular bisector line using point-slope form. (4) Solve the system of two equations to find the intersection point.

Method 2: Using the Circumradius Formula

The circumradius can be found using R = abc / (4K), where a, b, c are the side lengths and K is the area of the triangle. The area can be computed using the shoelace formula: K = |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)| / 2. Once R is known, the circumcenter coordinates can be verified by checking that the distance from the circumcenter to each vertex equals R.

Special Cases

  • Right Triangle: The circumcenter is always at the midpoint of the hypotenuse. The circumradius equals half the hypotenuse length.
  • Acute Triangle: The circumcenter lies strictly inside the triangle.
  • Obtuse Triangle: The circumcenter lies outside the triangle, on the opposite side from the obtuse angle.
  • Equilateral Triangle: The circumcenter, centroid, incenter, and orthocenter all coincide at the same point.
  • Isosceles Triangle: The circumcenter lies on the axis of symmetry (the perpendicular bisector of the base).

Circumcenter vs. Other Triangle Centers

  • Circumcenter: Center of the circumscribed circle, equidistant from all vertices.
  • Incenter: Center of the inscribed circle, equidistant from all sides.
  • Centroid: Intersection of medians, the center of mass of the triangle.
  • Orthocenter: Intersection of altitudes.

These four points are collinear only in an isosceles triangle (they all lie on the Euler line). In general, the circumcenter, centroid, and orthocenter are collinear on the Euler line, with the centroid dividing the segment from circumcenter to orthocenter in a 1:2 ratio.

Real-World Applications

  • Cell Tower Placement: Finding the circumcenter helps determine the optimal placement for a tower equidistant from three locations.
  • Triangulation: Used in surveying and GPS systems to determine positions from three reference points.
  • Computer Graphics: Delaunay triangulation and Voronoi diagrams use circumcenters extensively.
  • Astronomy: Determining the circular orbit passing through three observed positions of a celestial body.