Enter the three vertices of your triangle — Point A (x1, y1), Point B (x2, y2), and Point C (x3, y3) — and the Triangle Circumcenter Calculator finds the exact circumcenter coordinates (x, y) along with the circumradius. The circumcenter is the point equidistant from all three vertices, forming the center of the triangle's circumscribed circle.
Circumcenter X
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Circumcenter Y
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Circumradius (R)
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Triangle Type
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The circumcenter of a triangle is the point where all three perpendicular bisectors of the triangle's sides intersect. It is equidistant from all three vertices of the triangle and serves as the center of the circumscribed circle (circumcircle) that passes through all three vertices.
To find the circumcenter, calculate the perpendicular bisectors of at least two sides of the triangle and find their intersection point. For a triangle with vertices (x1,y1), (x2,y2), and (x3,y3), you set up two equations from the perpendicular bisectors and solve the resulting system of linear equations. This calculator does all that work for you automatically.
Yes, every non-degenerate triangle has exactly one circumcenter. The only exception is when the three points are collinear (lie on a straight line), in which case they do not form a valid triangle and no circumcenter exists. This calculator will alert you if your points are collinear.
For an acute triangle, the circumcenter lies inside the triangle. For a right triangle, the circumcenter is exactly at the midpoint of the hypotenuse. For an obtuse triangle, the circumcenter falls outside the triangle, on the opposite side of the longest edge.
For a right triangle, the circumcenter is simply the midpoint of the hypotenuse (the side opposite the right angle). The circumradius equals half the length of the hypotenuse. You can verify this with the calculator by entering a right triangle's vertices.
For an equilateral triangle, the circumcenter coincides with the centroid, incenter, and orthocenter — all at the same point. It is located at one-third of the height from the base, at the geometric center of the triangle. All three vertices are exactly one circumradius away from this center point.
The circumradius (R) is the radius of the circumscribed circle — the distance from the circumcenter to any of the three vertices. It can be calculated using the formula R = (a × b × c) / (4 × Area), where a, b, c are the side lengths and Area is the area of the triangle. Once the circumcenter coordinates are known, R is simply the distance from that point to any vertex.
To construct the circumcenter geometrically: (1) Draw the perpendicular bisector of one side by setting your compass to more than half the side's length, drawing arcs from both endpoints, and connecting the arc intersections. (2) Repeat for a second side. (3) The point where the two perpendicular bisectors meet is the circumcenter. A third bisector can be drawn as a check — it should pass through the same point.