Circumcenter Of A Triangle Calculator - Cx, Cy And Circumradius

The circumcenter of a triangle calculator returns the point equidistant from all three vertices of a triangle, given three pairs of coordinates. That point is the center of the unique circle that passes through the three vertices, also called the circumscribed circle, and the distance from the circumcenter to any vertex is the circumradius R. Type the six vertex coordinates in the same length unit and the page reports the (Cx, Cy) point, the circumradius, and a triangle type label in a single pass.

• • Coordinate geometry homework and exam problems: Verify textbook problems that hand you three vertex coordinates and ask for the circumcenter, the circumradius, or the equation of the circumscribed circle.
• • Plotting a circumscribed circle on a coordinate plane: Plot the (Cx, Cy) point as the circle's center and use the circumradius R to draw a circle through all three vertices.
• • Euler line and triangle center work: Compute the circumcenter as one of the four classical triangle centers and place it on the Euler line alongside the orthocenter and centroid.

A triangle in the plane is uniquely defined by three non-collinear points, so the circumcenter is a function of six coordinates. The output is in the same length unit as the input coordinates, so switch units before the calculation when the source and the report disagree.

For the same three vertex coordinates the Centroid Calculator returns the arithmetic mean of the three vertices, which is the second of the four classical triangle centers and lands inside the triangle for every non-degenerate case.

The circumcenter of a triangle is the point equidistant from all three vertices. Equating the squared distance from an unknown point (Cx, Cy) to vertex 1 and vertex 2 gives the perpendicular bisector of side 1-2, and equating the distance to vertex 1 and vertex 3 gives the perpendicular bisector of side 1-3. Solving the two-line intersection yields (Cx, Cy) in closed form.

• x1, y1, x2, y2, x3, y3: Cartesian coordinates of the three triangle vertices, in the same length unit. Reorder the vertices freely because the closed form is symmetric in the three inputs.
• Cx, Cy: Circumcenter coordinates returned by the perpendicular-bisector intersection, in the same length unit as the input coordinates.
• R: Circumradius, computed as the distance from the circumcenter to any vertex. R is the same in all three directions for a non-degenerate triangle.

The same closed form feeds the circumradius, computed as the distance from the circumcenter to any of the three vertices. For a non-degenerate triangle all three distances agree, and the page reports the result in the same length unit as the input coordinates.

The cross-check column on the right recomputes the circumcenter from a different pair of perpendicular bisectors (sides 1-2 and 2-3) and confirms the two intersections land on the same point for a valid triangle.

According to Wolfram MathWorld, the circumcenter of a triangle is the center of the unique circle passing through all three vertices, obtained as the intersection of the three perpendicular bisectors of the sides.

The same six coordinates also feed the orthocenter of the triangle, and the Orthocenter Calculator returns the altitude intersection (Hx, Hy) in the same length unit, so the two triangle centers can be compared side by side on the Euler line.

Four ideas decide what the circumcenter means for a triangle and how the closed form behaves on edge cases.

For any non-equilateral triangle the circumcenter, centroid, and orthocenter are collinear on the Euler line, with the centroid sitting one third of the way from the circumcenter to the orthocenter. The result is in the same length unit as the input coordinates, so the page does not perform any unit conversion.

For a right triangle the circumcenter coincides with the midpoint of the hypotenuse, and the Right Triangle Calculator explores the same right-triangle case with the leg lengths, the hypotenuse, and the right-angle vertex in the same length unit.

Use the circumcenter of a triangle calculator with all six coordinates in the same length unit, then read the (Cx, Cy) point, the circumradius R, and the triangle type label in a single pass.

1. 1 Enter the first vertex: Type the x and y coordinates of vertex 1 in the same length unit you will use for the other two vertices. The closed form is symmetric, so the order of the three vertices does not change the result.
2. 2 Enter the second and third vertices: Type the x and y coordinates of vertices 2 and 3 in the same length unit as vertex 1. Mixing units in the same calculation is the most common source of a wrong circumcenter.
3. 3 Read the circumcenter point and circumradius: The black results panel reports the formatted (Cx, Cy) point and the circumradius R as the primary answer, with Cx and Cy also listed as separate numeric rows.
4. 4 Cross-check with the second pair of bisectors: The right-hand column shows the circumcenter recomputed from the perpendicular bisectors of sides 1-2 and 2-3. The two values match for any non-degenerate triangle.
5. 5 Check the triangle type label: Use the triangle type label to confirm the inside, midpoint, or outside rule: acute puts the circumcenter inside the triangle, right at the midpoint of the hypotenuse, and obtuse outside.

For a triangle with vertices (0, 0), (6, 0), and (0, 9), the circumcenter is (3, 4.5). The result panel reports Cx = 3 and Cy = 4.5, the cross-check also returns (3, 4.5), the circumradius R = 5.4083, and the triangle type label reads 'Right triangle'.

When the three vertices form an acute triangle the circumcenter sits inside the figure, and the Acute Triangle Calculator explores the same acute case with the angle sum and the side length bounds that the inside-circumcenter rule depends on.

The closed form is short, but the circumcenter of a triangle calculator wraps it in a layout that helps with plotting, verifying, and reporting the circumcenter of a triangle.

• • Direct from six coordinates: The closed form takes the three vertices straight to the circumcenter, so no side lengths, area, or perpendicular heights are needed first.
• • Circumcenter point is shown explicitly: The page formats the circumcenter as (Cx, Cy) so it can be plotted on a coordinate plane or pasted into a report without further formatting.
• • Cross-check with a second pair of perpendicular bisectors: The cross-check column recomputes the circumcenter from the perpendicular bisectors of sides 1-2 and 2-3 in the same pass.
• • Circumradius is included: The circumradius R is reported as a numeric value in the same length unit as the inputs, so the page supports both plotting the center and drawing the circle with one calculation.
• • Triangle type label is built in: The page reports the triangle as acute, right, obtuse, or degenerate, so the user knows immediately whether the circumcenter is inside, on the hypotenuse, outside, or not defined.

The circumcenter of a triangle calculator returns the (Cx, Cy) point, the circumradius R, the cross-check, and the triangle type label together, which is enough data to plot the circumscribed circle without a second pass.

When the input comes from a triangle stored as three vertex coordinates, the Centroid Of A Triangle Calculator returns the centroid of the same triangle in the same length unit, so the circumcenter and the centroid can be plotted together on the same coordinate plane.

The closed form is stable, but a few input choices decide whether the circumcenter of a triangle calculator returns the right value for a given triangle.

• • The calculator does not solve for a missing vertex when only the circumcenter and two vertices are known. Use a system of equations for that case.
• • Hand calculation that rounds the intermediate t, u, or J partway through can differ by a few thousandths of a length unit from the calculator, which keeps full precision until the display step.
• • For polygons with more than three sides the circumcenter is not the right tool; the circumscribed circle of a polygon is a different problem.

For an obtuse triangle the circumcenter sits outside the triangle, so a plot on a coordinate plane will need extra room around the triangle. For an equilateral triangle the circumcenter coincides with the centroid, orthocenter, and incenter.

According to Wikipedia, the circumcenter of a triangle lies inside an acute triangle, at the midpoint of the hypotenuse for a right triangle, and outside an obtuse triangle.

The same six coordinates that produce the circumcenter also produce the triangle area through the shoelace formula, and the Area Triangle Coordinates Calculator is the natural next step when both the center and the area of a triangle are needed in the same report.