Circumscribed Circle Calculator

A circumscribed circle, usually called a circumcircle, is a circle that passes through all three vertices of a triangle. Its center is the circumcenter, the point where the three perpendicular bisectors of the triangle sides meet. Because every vertex lies the same distance from that point, this shared distance is the circumradius. A unique circumcircle exists for every non-degenerate triangle, whether the triangle is acute, right, or obtuse. This calculator works directly from Cartesian coordinates. It substitutes the three points into the general circle equation and solves the resulting linear differences for the center coordinates. The determinant in that calculation is twice the triangle's signed coordinate expression. If the determinant is zero, the points lie on one line and cannot define a finite circle. Otherwise, the radius is the Euclidean distance from the computed center to any vertex, and the area is pi multiplied by the radius squared. The circumcenter's location reveals useful geometry. For an acute triangle it lies inside the triangle. For a right triangle it is exactly the midpoint of the hypotenuse, which makes the hypotenuse the circle's diameter. For an obtuse triangle the center lies outside the triangle. These facts provide quick visual checks on a computed result and connect coordinate algebra with classical straightedge-and-compass construction. Coordinate order does not affect the final circle. Swapping vertices may reverse the determinant's sign, but the center and radius remain unchanged. Decimal and negative coordinates are accepted, so the tool can handle translated or scaled diagrams as well as simple textbook triangles. Results are rounded only for display; calculations retain JavaScript double-precision values internally. Circumcircles appear in geometry proofs, surveying, computer graphics, mesh generation, navigation, and engineering design. The center can identify an equal-distance facility location, while the circle equation supports collision tests and geometric modeling. For very large coordinates or measurements requiring certified tolerances, retain appropriate units and verify rounding independently. For ordinary classroom and design problems, entering three measured vertices gives an immediate, transparent description of the circle that contains them.