Circumcenter Calculator (Circle Through Three Points)

How it works

D = 2[x_A(y_B − y_C) + x_B(y_C − y_A) + x_C(y_A − y_B)] · U_x = [(x_A² + y_A²)(y_B − y_C) + (x_B² + y_B²)(y_C − y_A) + (x_C² + y_C²)(y_A − y_B)] / D · U_y = [(x_A² + y_A²)(x_C − x_B) + (x_B² + y_B²)(x_A − x_C) + (x_C² + y_C²)(x_B − x_A)] / D · R = |UA|

The circumcenter is the point where the perpendicular bisectors of the three sides meet. It is equally far from all three vertices, so it is the center of the circumscribed circle — the unique circle through three non-collinear points. It lies inside an acute triangle, at the midpoint of the hypotenuse of a right triangle and outside an obtuse triangle. Exact fractions are used throughout.

Worked example

A(0, 0), B(6, 0), C(2, 4): the bisector of AB is x = 3; the bisector of AC passes through (1, 2) with slope −1/2, so at x = 3 it gives y = 1. The circumcenter is (3, 1), the radius is √(3² + 1²) = √10 ≈ 3.162278, and the circle is (x − 3)² + (y − 1)² = 10. The triangle is acute, so the center is inside.

Sources: Wolfram MathWorld — Circumcenter · Khan Academy — Circumcenter of a triangle

Frequently asked questions