Math

Instant, private, and free

Circumcenter of a Triangle Calculator.

Find the circumcenter and circumradius of a triangle from three Cartesian vertices.

On-device calculationNo signup
01

Set your values

Results update as you type.

Center x: 2 units

Center x

0.000000units
Center y: 1.5 units

Center y

0.000000units
Radius: 2.5 units

Radius

0.000000units

Results update automatically as you type.

Use Cases

Geometry homework and classwork

Quickly verify the circumcenter and circumradius for triangles given in coordinate geometry problems, saving time on manual calculations.

Example: Check the circumcenter of a triangle with vertices (0,0), (4,0), and (0,3).

Design and engineering applications

Find the center of a circle that passes through three known points, useful in CAD, robotics, or any field requiring a circle fit to three points.

Example: Determine the center of a circular plate that must pass through three mounting holes.

Frequently Asked Questions

What is the circumcenter of a triangle?
The circumcenter is the point equidistant from all three vertices of a triangle. It is the center of the circle that passes through all three vertices, known as the circumcircle. The distance from the circumcenter to any vertex is the circumradius.
How is the circumcenter calculated from the coordinates?
Given the coordinates of the three vertices (x1,y1), (x2,y2), and (x3,y3), the circumcenter is found by solving the perpendicular bisectors of two sides. The calculator uses these coordinates to compute the intersection point, which is the circumcenter, and then calculates the distance to any vertex to get the circumradius.
Can the circumcenter be outside the triangle?
Yes. For an obtuse triangle, the circumcenter lies outside the triangle. For a right triangle, it lies on the hypotenuse midpoint. For an acute triangle, it lies inside. The calculator will output the coordinates regardless of the triangle type.

Tips & Common Mistakes

Tips

  • Ensure the three points are not collinear; otherwise, the circumcenter is undefined and the calculator may not give a valid result.
  • Double-check the signs of your coordinates, especially if some are negative, to avoid input errors.
  • Use the circumradius to verify the circumcenter: the distance from the center to each vertex should be the same.
  • For a right triangle, the circumcenter is the midpoint of the hypotenuse, which can be a quick sanity check.

Common Mistakes to Avoid

  • Entering the coordinates in the wrong order, such as swapping x and y for a point, which will give an incorrect circumcenter.
  • Using decimal approximations for coordinates that are exact fractions, which may lead to slight inaccuracies in the result.
  • Assuming the circumcenter is always inside the triangle; it can be outside for obtuse triangles.

Last updated: August 13, 2026