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 circumscribed circle (circumcircle) that passes through all three vertices.

Where does the circumcenter lie relative to the triangle?

For an acute triangle the circumcenter is inside the triangle. For a right triangle it lies on the hypotenuse midpoint. For an obtuse triangle it falls outside the triangle.

What happens if the three points are collinear?

Collinear points do not form a triangle, so no circumcircle exists. The calculator will detect this and display an error instead of producing a result.

How is the circumradius calculated?

After finding the circumcenter, the circumradius R is simply the Euclidean distance from the circumcenter to any one of the three vertices.

Is this the same as the centroid?

No. The centroid is the average of the three vertices (center of mass). The circumcenter is equidistant from the vertices. They only coincide for equilateral triangles.

Triangle Circumcenter Calculator

Name: Triangle Circumcenter Calculator
Availability: InStock
Author: Nham Vu

Enter the three vertices of a triangle to find the circumcenter (circumscribed circle center) and circumradius.

Vertex Coordinates

Vertex A

Vertex B

Vertex C

Quick Presets

Enter vertex coordinates and click Calculate

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Summary

Enter the three vertices of a triangle to find the circumcenter (circumscribed circle center) and circumradius.

How it works

Enter the x and y coordinates for each of the three triangle vertices.
The calculator solves the system of perpendicular bisector equations to find the circumcenter.
The circumradius is computed as the distance from the circumcenter to any vertex.
Results show the circumcenter coordinates and the equation of the circumscribed circle.

Use cases

Finding the center of the circle that passes through all three vertices of a triangle.
Solving computational geometry problems in game development or graphics.
Checking geometry homework or exam answers for circumscribed circle problems.
Engineering and CAD work requiring a circle through three known points.
Teaching or studying the relationship between triangle type and circumcenter position.

Frequently Asked Questions

Last updated: 2026-07-04 · Reviewed by Nham Vu