Goal
Find the Cartesian circumcenter and circumradius of a non-collinear triangle from its three ordered vertices.
Circumcenter of a 2D Triangle
Find the Cartesian circumcenter and circumradius of a non-collinear triangle from its three ordered vertices.
Key facts
- What it does
- Find the Cartesian circumcenter and circumradius of a non-collinear triangle from its three ordered vertices.
- Formula
For u=B-A, v=C-A, D=u_x v_y-u_y v_x, O=A+((|u|^2 v_y-|v|^2 u_y)/(2D), (u_x|v|^2-v_x|u|^2)/(2D)).- You enter
- Vertex A x1 · Vertex A y1 · Vertex B x2 · Vertex B y2 · Vertex C x3 · Vertex C y3
- Worked example
- The circumcenter is (2, 1.5) and the circumradius is 2.5.
A clearer path to an answer
From your question to a useful result
This page keeps the calculation transparent: define the goal, enter the matching values, inspect the method, and decide what the result means in your situation.
Inputs
Vertex A x1 · Vertex A y1 · Vertex B x2 · Vertex B y2 · Vertex C x3 · Vertex C y3
Method
For u=B-A, v=C-A, D=u_x v_y-u_y v_x, O=A+((|u|^2 v_y-|v|^2 u_y)/(2D), (u_x|v|^2-v_x|u|^2)/(2D)).
Next step
Calculate, review the assumptions below, then compare a related tool when the decision needs more context.
Circumcenter of a 2D Triangle
Find the Cartesian circumcenter and circumradius of a non-collinear triangle from its three ordered vertices.
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Result
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01
Inputs (6)
- Vertex A x1 Ready
- Vertex A y1 Ready
- Vertex B x2 Ready
- Vertex B y2 Ready
- +2 more inputs
02
Formula
For u=B-A, v=C-A, D=u_x v_y-u_y v_x, O=A+((|u|^2 v_y-|v|^2 u_y)/(2D), (u_x|v|^2-v_x|u|^2)/(2D)).
Bounded, transparent calculation
03
Result
- Calculate to preview the result.
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Formula, assumptions, and example
Formula: For u=B-A, v=C-A, D=u_x v_y-u_y v_x, O=A+((|u|^2 v_y-|v|^2 u_y)/(2D), (u_x|v|^2-v_x|u|^2)/(2D)).
The circumcenter is the point equidistant from the three vertices. The calculator translates the triangle to vertex A, solves the perpendicular-bisector equations, and returns the center and common radius.
- The three vertices are finite points in one two-dimensional Cartesian coordinate system and are not collinear or nearly collinear.
- The result is the Euclidean circumcenter and circumradius of the triangle; no spherical geometry, unit conversion, or finite-drawing interpretation is added.
Worked example: The circumcenter is (2, 1.5) and the circumradius is 2.5.
Displayed input contract
- Vertex A x1 · minimum -1000000 · maximum 1000000
- Vertex A y1 · minimum -1000000 · maximum 1000000
- Vertex B x2 · minimum -1000000 · maximum 1000000
- Vertex B y2 · minimum -1000000 · maximum 1000000
- Vertex C x3 · minimum -1000000 · maximum 1000000
- Vertex C y3 · minimum -1000000 · maximum 1000000
The displayed limits are checked before the handler runs. Model-specific domain checks may also reject impossible or non-finite inputs.
Methodology: This calculator follows the WorldCalculate input, formula, precision, and boundary policy. Read the official methodology.
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Answer-first guide
How to use the Circumcenter of a 2D Triangle for a real question
Find the Cartesian circumcenter and circumradius of a non-collinear triangle from its three ordered vertices. Start with one clearly defined goal, enter values in the units shown, and keep the result attached to the assumptions below.
What this answers
This tool is useful when your question includes circumcenter of triangle, triangle circumcenter, circumcircle center. It returns the outputs declared in the calculator contract rather than a live quote, approval, diagnosis, or professional sign-off.
What you enter
Vertex A x1 · Vertex A y1 · Vertex B x2 · Vertex B y2 · Vertex C x3 · Vertex C y3. Keep the same time period, unit system, and currency wherever the form requires comparable values.
How to check it
Run the worked example first, compare its output with the page's example, then change one input at a time. This makes an unexpected result easier to trace to a unit, boundary, or assumption.
Three checks before you rely on the answer
- Match the question. Confirm that the result means the quantity you need, not a similar-sounding percentage, balance, rate, or estimate.
- Match the inputs. Use the requested units and period, and read each hint before replacing the example values with your own.
- Read the boundary. Review the assumptions and limits. The three vertices are finite points in one two-dimensional Cartesian coordinate system and are not collinear or nearly collinear.
Need a wider view? Browse Math Calculators or compare the related tools below. The WorldCalculate methodology explains how formulas, examples, limits, and revisions are reviewed.
How to use the Circumcenter of a 2D Triangle
- Enter Vertex A x1 — The x coordinate of vertex A.
- Enter Vertex A y1 — The y coordinate of vertex A.
- Enter Vertex B x2 — The x coordinate of vertex B.
- Enter Vertex B y2 — The y coordinate of vertex B.
- Enter Vertex C x3 — The x coordinate of vertex C.
- Enter Vertex C y3 — The y coordinate of vertex C.
- Choose Calculate and read the result panel.
- Use Download PDF or Download Word to save a result sheet.
Formula
For u=B-A, v=C-A, D=u_x v_y-u_y v_x, O=A+((|u|^2 v_y-|v|^2 u_y)/(2D), (u_x|v|^2-v_x|u|^2)/(2D)).
The circumcenter is the point equidistant from the three vertices. The calculator translates the triangle to vertex A, solves the perpendicular-bisector equations, and returns the center and common radius.
Worked example
The circumcenter is (2, 1.5) and the circumradius is 2.5.
Assumptions and limits
- The three vertices are finite points in one two-dimensional Cartesian coordinate system and are not collinear or nearly collinear.
- The result is the Euclidean circumcenter and circumradius of the triangle; no spherical geometry, unit conversion, or finite-drawing interpretation is added.
Who uses this calculator?
- Analytic-geometry and trigonometry students
- Graphics, surveying, and engineering learners
- Developers working with triangle geometry
When is it useful?
- Find the center and radius of a circle through three vertices.
- Check perpendicular-bisector and determinant calculations.
- Prepare triangle center data for a separate orthocenter or geometry workflow.
Context and background
The mathematical structure behind the tools
Math calculators move from named quantities to a relation, then to a result that can be checked with substitution, units, or an alternate form.
Arithmetic, algebra, geometry, trigonometry, and number theory provide reusable structures for classroom work and everyday reasoning. Each page narrows that structure to one declared problem.
Research and review
How this guide was researched
Researched by Hassan ALRowaie, Founder and editorial researcher at WorldCalculate.
This guide follows the live calculator's declared inputs, formula, worked example, assumptions, validation boundaries, and source-backed methodology. The review date describes editorial review of the calculator explanation; it is not a promise that external facts or rates remain current.
The circumcenter of a triangle is the point that has equal Euclidean distance from all three vertices. This calculator accepts the ordered Cartesian coordinates of vertices A, B, and C and returns the center coordinates together with the common circumradius. It uses translated edge vectors, a signed two-dimensional determinant, and the equations of the perpendicular bisectors. A non-collinear triangle is required; a collinear or nearly collinear set has no stable finite circumcenter under this contract. The page is limited to a flat two-dimensional triangle and does not fit noisy points, convert units, or interpret a drawn finite object. This guide explains the fields, derivation, examples, degeneracy rule, checks, precision, and safe reuse.
What the circumcenter result means
The returned center is a point O = (Ox, Oy) in the same Cartesian coordinate system as the three vertices. The returned radius is the distance from O to A, B, or C. In exact Euclidean geometry those three distances are equal because the center lies at the intersection of the triangle's perpendicular bisectors. The calculator reports the center and radius as numeric values rather than drawing a circle or returning a collection of points.
The vertices are ordered as A, B, and C for a reproducible determinant calculation. Reordering the vertices changes the orientation sign of the translated determinant in some permutations, but it does not change the geometric center when the same three points are used. The order still matters for reading the form and for preserving a clear source record. Do not exchange one x coordinate without its paired y coordinate.
The circumcenter can lie inside, on, or outside the triangle. An acute triangle has an interior circumcenter, a right triangle has its center at the midpoint of the hypotenuse, and an obtuse triangle has a center outside the triangular region. Being outside the triangle does not mean the circle calculation failed. It reflects the geometry of the three vertices.
The result assumes ordinary Euclidean distance in a flat plane. If the coordinates are longitude and latitude, points on a curved surface, or points in a distorted drawing, the returned center is a Cartesian construction and may not be the appropriate geographic or physical center. Establish the coordinate model before using the numbers.
- The output is a Cartesian center and one common Euclidean radius.
- The center is equidistant from all three ordered vertices.
- The center may be outside an obtuse triangle and still be correct.
- The model is flat two-dimensional Cartesian geometry.
The six vertex fields
The first pair x1, y1 defines vertex A. The second pair x2, y2 defines B. The third pair x3, y3 defines C. Each x coordinate belongs to the y coordinate with the same suffix. The labels use vertex names as well as coordinate suffixes so that a point is not accidentally treated as two unrelated scalar inputs.
Every coordinate accepts a finite number from negative 1,000,000 through positive 1,000,000. Negative coordinates, zeros, and fractional coordinates are valid. The visible bounds and the pure handler agree. A blank, non-finite, string, or out-of-range value is rejected before edge differences and squared lengths are calculated.
The coordinates should use one origin, axis orientation, and length convention. Translating every vertex by the same vector translates the center by that vector and leaves the radius unchanged. Scaling coordinates changes both center coordinates and radius by the scale factor in the appropriate direction. The calculator does not perform these transformations; they are useful properties for checking a source.
Three distinct labels do not guarantee three distinct geometric points. If two vertices coincide, or if all three lie on one line, the determinant test identifies a degenerate or unstable triangle. The handler does not invent a center from a segment or return a midpoint as a fallback because that would answer a different question.
- Keep each x coordinate paired with its same-suffix y coordinate.
- All six coordinates use inclusive finite signed bounds.
- Use one Cartesian origin, orientation, and length convention.
- Distinct field names do not guarantee a non-degenerate triangle.
Translated edges and the determinant
The calculation translates vertex A to the origin conceptually. Define u = B - A = (x2 - x1, y2 - y1) and v = C - A = (x3 - x1, y3 - y1). These two edge vectors contain the triangle's shape relative to A. The signed determinant D = u_x v_y - u_y v_x is twice the signed area of the triangle. Its sign records the orientation of A, B, and C, while its absolute value measures area scale.
If D is zero, u and v are linearly dependent. The three vertices are collinear or a vertex is repeated, and the perpendicular bisectors do not meet at one finite unique point. The handler rejects this case. It also rejects a determinant whose absolute value is no larger than a scale-aware tolerance, because a very thin triangle can produce unstable divisions even when its determinant is not exactly zero.
The scale used by the guard is based on the largest absolute translated edge component, with a minimum scale of one. The tolerance is 1e-12 times the square of that scale. This makes the nearly-collinear rule respond to coordinate magnitude rather than using one absolute number for every triangle. It is a software stability boundary, not a universal geometric definition of when a triangle is visually thin.
The determinant is used in the denominator as 2D. A clockwise and counterclockwise ordering gives opposite D values, but the offset formulas change sign consistently and return the same center. Keeping D signed avoids an unnecessary absolute-value transformation and makes the orientation behavior explicit for review.
- u and v are the edges from A to B and A to C.
- D is the signed cross product and twice the signed triangle area.
- Zero or scale-small D means the circumcenter is rejected as unstable.
- Vertex orientation changes the determinant sign, not the geometric center.
Solving the perpendicular-bisector equations
A point O is equidistant from A and B when |O - A| squared equals |O - B| squared. Expanding and canceling the common O-squared terms produces a linear equation involving the edge u. The same process for A and C produces a second equation involving v. Solving those two equations gives the offset from A to the circumcenter. The calculator uses this translated two-equation form rather than repeatedly expanding three full circle equations.
Let U2 = u_x squared + u_y squared and V2 = v_x squared + v_y squared. The x offset is (U2 v_y - V2 u_y)/(2D). The y offset is (u_x V2 - v_x U2)/(2D). Adding those offsets to x1 and y1 gives the center coordinates. The radius is then the Euclidean distance from that center back to A. Every derived value is checked for finiteness.
For A = (0, 0), B = (4, 0), and C = (0, 3), u = (4, 0), v = (0, 3), D = 12, U2 = 16, and V2 = 9. The x offset is 48/24 = 2 and the y offset is 36/24 = 1.5. The center is (2, 1.5), and its distance to A is sqrt(4 + 2.25) = 2.5. The same radius reaches B and C.
The formula does not assume that one edge is horizontal or that the triangle contains the origin. Translation by A handles arbitrary coordinates, while the determinant handles arbitrary orientation. The right-triangle example is convenient for arithmetic, not a hidden restriction of the handler.
- Equal-distance equations become two linear bisector equations.
- Solve offsets from A, then translate back to the original coordinates.
- The radius is computed from the returned center to a vertex.
- No horizontal-edge or origin assumption is required.
Worked acute, right, and obtuse examples
For the right triangle A = (0, 0), B = (4, 0), C = (0, 3), the midpoint of the hypotenuse from B to C is (2, 1.5). The calculator returns that point as the circumcenter and 2.5 as the radius. The distance to A is 2.5, the distance to B is 2.5, and the distance to C is 2.5. This is a simple independent check of equal radii.
For an acute isosceles triangle A = (-2, 0), B = (2, 0), C = (0, 3), symmetry places the center on the y-axis. The perpendicular bisector of AB is x = 0. Equal-distance comparison with A and C gives a center y coordinate determined by the two squared distances. The result is (0, 5/6) and a radius of 13/6, approximately 2.166667. Symmetry provides a useful check even if the displayed decimals are rounded.
For an obtuse triangle such as A = (0, 0), B = (6, 0), and C = (1, 1), the center lies outside the triangle. The calculator still solves the same bisector equations. A center outside the interior is expected when one angle is obtuse because the associated circumcircle must pass through all three vertices. Do not reject a result merely because it is not inside the triangular region.
A translated example can check coordinate invariance. Add (10, -4) to every vertex in a known triangle. The new center should be the old center plus (10, -4), while the radius should remain the same. This test checks that the final center adds the original A coordinate after solving offsets and that the radius uses center-to-vertex differences.
- A right triangle's center is the hypotenuse midpoint.
- Symmetry can predict a center coordinate in an isosceles triangle.
- An obtuse triangle may have an outside circumcenter.
- Translating all vertices translates the center but preserves radius.
Orientation, scaling, and geometric invariants
The center is independent of clockwise versus counterclockwise vertex order. Swapping B and C reverses the determinant and changes the signs of the numerator terms consistently. A robust check can calculate both orderings and compare the center and radius within a numerical tolerance. A disagreement larger than rounding indicates a component pairing or formula error.
Uniform scaling about the origin multiplies every center coordinate and the radius by the same positive scale. Scaling about another point combines scaling with translation. Non-uniformly scaling x and y generally changes angles and circle geometry, so there is no simple radius invariant. The calculator assumes the supplied coordinates already have the intended metric and does not rescale one axis.
The center is equidistant from all vertices, and the radius is nonnegative. These are the main output invariants. The center need not be the centroid, incenter, or orthocenter. Those are different triangle centers with different definitions. Naming the requested center precisely prevents a correct circumcenter from being mistaken for another average or intersection point.
A circle through three non-collinear points is unique in the Euclidean plane. The determinant condition is therefore both an algebraic denominator check and a geometric uniqueness check. When the triangle becomes very thin, the unique mathematical center can move far away and become highly sensitive to small coordinate changes. The near-collinearity guard keeps that unstable regime outside the K1 result contract.
- Vertex reordering should preserve center and radius.
- Uniform scaling preserves shape and scales center and radius predictably.
- Circumcenter is distinct from centroid, incenter, and orthocenter.
- Near-collinear triangles can have extreme center sensitivity.
Degenerate triangles and numerical precision
A repeated vertex gives an edge of zero length and a determinant of zero. Three collinear vertices also give zero determinant. In either case, there is no unique circle with a finite center through the intended triangle in the ordinary construction. The handler throws a clear validation error rather than returning infinity, NaN, a segment midpoint, or an arbitrary large center.
A nearly collinear triangle can have a small nonzero determinant. Dividing by 2D then amplifies small changes in edge squares and coordinates. The handler compares the determinant with a scale-aware tolerance before division. This policy is intentionally conservative for renderer-safe behavior. It does not claim that every application should use exactly the same tolerance for surveying, graphics, or scientific data.
All coordinate inputs are bounded, but translated differences can be larger than one input coordinate and squared edge lengths can be larger again. The engine checks the determinant, squared edges, offsets, center coordinates, and radius for finiteness. This layered approach prevents an apparently valid input set from returning a non-renderable result if a future change alters the arithmetic path.
For measured coordinates, numerical conditioning and measurement uncertainty are related but not identical. A triangle can pass the software tolerance while its center remains sensitive to the precision of the source points. If the radius or center is used for a threshold, fit, or physical decision, assess uncertainty and sensitivity outside this calculator.
- Repeated or collinear vertices are invalid triangle geometry.
- A scale-aware determinant guard rejects unstable near-collinear cases.
- Derived edge, center, and radius values are checked for finiteness.
- Measurement uncertainty may require stricter review than the software guard.
Common mistakes and independent checks
Common mistakes include using the centroid formula, averaging the three vertices, or taking the midpoint of an arbitrary edge. Those operations do not generally produce the circumcenter. Another error is pairing x2 with y3 or forgetting to translate the solved offset back by A. Keep the three vertex pairs visible and use an equal-distance check after calculating.
For an independent check, calculate rA = distance(O, A), rB = distance(O, B), and rC = distance(O, C). The three values should agree within a tolerance. Also check that the center satisfies the two perpendicular-bisector equalities in squared-distance form. The radius returned by the handler is based on A, so comparing it with B and C tests the whole construction rather than only the last square root.
Test simple shapes: an axis-aligned right triangle, a symmetric isosceles triangle, a translated copy, a reversed vertex order, and a deliberately collinear triple. Include negative and fractional coordinates. These cases cover orientation, translation, output scale, pair ordering, and the invalid determinant branch without relying on one default answer.
If another source returns a different center, check whether it uses a different coordinate metric, rounds intermediate values, swaps a vertex coordinate, or solves a least-squares circle through noisy points. The K1 handler solves the exact three-point Euclidean problem and rejects unstable input; it is not a noisy circle-fitting service.
- Do not substitute centroid or edge-midpoint formulas.
- Verify equal distances from the returned center to all three vertices.
- Use shape, translation, order, sign, and degeneracy fixtures.
- Distinguish exact three-point geometry from noisy circle fitting.
Related triangle centers and downstream use
The circumcenter and circumradius describe the unique circle through the triangle's vertices. The orthocenter is a different center formed by the triangle's altitudes. In a triangle, the centroid comes from vertex averages and the incenter comes from angle-bisector geometry. The calculator returns only the circumcenter so the formula and validation remain explicit. Use a dedicated operation for another center rather than reinterpreting this output.
A circumcenter can support a circumcircle drawing, chord calculation, arc construction, or a later triangle-center identity. A drawing system may need to clip the infinite mathematical circle to a viewport, and a geometry engine may need a tolerance for point-on-circle tests. Those are downstream rules. The returned radius is not a pixel size, rendering command, or collision tolerance by itself.
If coordinates represent a physical survey or a design, retain the coordinate units and reference frame with the center. A circle computed in a local Cartesian patch may not be a geodesic circle on a curved surface. Likewise, a distorted image coordinate system can make the Euclidean circle differ from a circle in the underlying object. The page does not detect those conditions.
A reproducible handoff contains all six coordinates, vertex order, determinant status, center, radius, and the equal-distance tolerance used in review. This lets another person distinguish a formula result from a later visual or physical interpretation and makes it possible to diagnose a sensitive near-degenerate input.
- Circumcenter is one of several distinct triangle centers.
- Downstream drawings and point-on-circle tests need their own tolerances.
- Coordinate units and reference frames remain part of the result context.
- Store vertex order, determinant status, center, radius, and review tolerance.
Limits, privacy, and safety boundaries
This is a bounded exact three-point Euclidean construction. It does not calculate a best-fit circle from many samples, handle weighted points, use a curved-surface metric, repair collinearity, or infer a finite object boundary. It assumes the vertices are intentional and the coordinate system is appropriate. A finite numeric center does not prove that the source measurements are accurate or that a circle is the right model.
Vertex coordinates can reveal locations, floor plans, movement traces, medical geometry, or proprietary designs. Enter only the six values required by the formula and avoid names, account details, credentials, or raw records in shared notes. Protect the source data according to its sensitivity. The page's local arithmetic and small input count do not remove spatial privacy concerns.
Do not use an unreviewed circumcenter or radius as a navigation boundary, structural clearance, medical boundary, robotics command, or other high-consequence result. Validate the coordinate frame, units, measurement uncertainty, degeneracy margin, and downstream object limits with domain-specific tools and qualified reviewers.
If the triangle is derived from noisy observations, record the source precision and sensitivity of the center. A small change to a thin triangle can move the center substantially. The calculator's near-collinearity check is a safe software boundary, not a substitute for a statistical uncertainty analysis.
- The tool solves an exact flat three-point problem, not a noisy fit.
- Protect coordinates that may expose locations or designs.
- Validate units, uncertainty, object limits, and degeneracy before consequential use.
- Treat the determinant guard and measurement sensitivity as separate concerns.
Frequently asked questions
Can the circumcenter be outside the triangle? Yes. That is expected for an obtuse triangle. The defining property is equal distance to all three vertices, not inclusion inside the triangular region. For an acute triangle the center is inside, and for a right triangle it is the midpoint of the hypotenuse.
Why reject collinear points? A line does not have one finite circle center determined by three distinct points. The perpendicular bisectors are parallel or fail to meet uniquely, and a repeated point gives the same degeneracy. The handler also rejects nearly collinear triangles because the mathematical answer becomes numerically unstable even before the determinant reaches zero.
Is the circumcenter the average of the vertices? No. The average is the centroid. The circumcenter is obtained by intersecting perpendicular bisectors and generally has different coordinates. A symmetric example can make the two centers share one coordinate, but their definitions and results are not interchangeable.
Does changing vertex order change the answer? Reordering the same three point pairs should preserve the geometric center and radius. It can reverse the determinant sign, so a test should compare final outputs rather than the signed intermediate determinant. Changing a coordinate without its paired coordinate is not a reorder; it changes the triangle.
- Outside-the-triangle centers are valid for obtuse triangles.
- Collinearity removes the unique finite circumcircle center.
- The centroid and circumcenter are different constructions.
- Reordering point pairs should preserve the final center and radius.
Frequently asked questions
What is the Circumcenter of a 2D Triangle?
Find the Cartesian circumcenter and circumradius of a non-collinear triangle from its three ordered vertices.
What is the formula for the Circumcenter of a 2D Triangle?
For u=B-A, v=C-A, D=u_x v_y-u_y v_x, O=A+((|u|^2 v_y-|v|^2 u_y)/(2D), (u_x|v|^2-v_x|u|^2)/(2D)). The circumcenter is the point equidistant from the three vertices. The calculator translates the triangle to vertex A, solves the perpendicular-bisector equations, and returns the center and common radius.
What do I need to use this calculator?
Enter Vertex A x1, Vertex A y1, Vertex B x2, Vertex B y2, Vertex C x3, Vertex C y3, then choose Calculate.
What are the limits of this calculator?
The three vertices are finite points in one two-dimensional Cartesian coordinate system and are not collinear or nearly collinear. The result is the Euclidean circumcenter and circumradius of the triangle; no spherical geometry, unit conversion, or finite-drawing interpretation is added.
Methodology
This calculator is part of the WorldCalculate library. Its formula, example, assumptions, input bounds, and output formatting follow the official methodology.
Use this calculator as part of a bigger plan
These WorldCalculate collections connect this tool with related questions while keeping each calculation separate and transparent.