Circumcenter Calculator

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Circumcenter

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Table of Contents

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What is Circumcenter?

The circumcenter of a triangle is the point equally distant from its three vertices. It is also the center of the circle passing through those vertices. This calculator finds that point from three pairs of Cartesian coordinates and reports its horizontal and vertical coordinates.

The circumcenter is not generally the average of the vertices. That average gives the centroid, a different triangle center. The circumcenter instead comes from equal-distance conditions, or equivalently the intersection of perpendicular bisectors. Its position can be inside the triangle, on a side, or outside it.

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How to Use This Calculator

Enter each vertex as an x coordinate followed by its matching y coordinate. Keep the six values grouped into three pairs: A = (x₁, y₁), B = (x₂, y₂), and C = (x₃, y₃). Negative coordinates and decimal coordinates are valid.

Use one coordinate system with the same length scale on both axes. Click Calculate to obtain the circumcenter coordinates, rounded to three decimal places. The calculator reports the center, not the circumradius; you can find the radius afterward using the distance from that center to any vertex.

Before entering the values, check that the three points are distinct and do not lie on one straight line. Those cases do not define a unique circumcircle. The calculator rejects a zero denominator with a message that the points are collinear.

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Formula Used

Let S₁ = x₁² + y₁², S₂ = x₂² + y₂², and S₃ = x₃² + y₃². The shared denominator is D = 2[x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)].

The horizontal coordinate is Uₓ = [S₁(y₂ − y₃) + S₂(y₃ − y₁) + S₃(y₁ − y₂)]/D. The vertical coordinate is Uᵧ = [S₁(x₃ − x₂) + S₂(x₁ − x₃) + S₃(x₂ − x₁)]/D.

These formulas solve the equal-distance equations. Subtracting one squared-distance equation from another cancels the center's squared terms, leaving linear equations in Uₓ and Uᵧ. A zero D means those equations cannot locate one unique center.

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Variables Explained

VariableMeaning in this calculator
A = (x₁, y₁)The first vertex, entered as a horizontal coordinate and a vertical coordinate.
B = (x₂, y₂)The second vertex in the same coordinate system and unit.
C = (x₃, y₃)The third vertex. It must not lie on the line through the first two.
DA denominator proportional to the triangle's signed area. Zero indicates a degenerate triangle.
U = (Uₓ, Uᵧ)The circumcenter. Both coordinates are rounded to three decimal places in the output.
Circumradius, RThe distance from U to any vertex, such as √[(Uₓ − x₁)² + (Uᵧ − y₁)²]. It is not returned separately.
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Worked Example

Take A = (0, 0), B = (4, 0), and C = (0, 4). The shared denominator is 2[0(0 − 4) + 4(4 − 0) + 0(0 − 0)] = 32. The squared-coordinate sums are 0, 16 and 16.

The numerator for Uₓ is 0 + 16 × 4 + 0 = 64, giving Uₓ = 2. The numerator for Uᵧ is also 64, giving Uᵧ = 2. The result is therefore (2, 2).

Check it by measuring squared distances: to A, the squared distance is 2² + 2² = 8; to B, it is (2 − 4)² + 2² = 8; and to C, it is 2² + (2 − 4)² = 8. All three distances are √8, approximately 2.8284. Since this is a right triangle, the center is also the midpoint of its hypotenuse.

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Why Use This Calculator?

An outside result is not automatically an error. An obtuse triangle has its circumcenter outside the triangle; an acute triangle has it inside. A right triangle places it at the midpoint of the hypotenuse. The equal-distance check is more reliable than assuming every triangle center must be inside.

For plotting, mark the returned point using the same axis scale as the vertices. A graph stretched differently in the horizontal and vertical directions can make a true circle appear elliptical even when the coordinate calculation is correct.

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Common Mistakes

    Do not swap just one vertex's x and y values, confuse the circumcenter with the centroid, or mix pixels on one axis with centimeters on the other. Swapping the order of the complete vertex pairs leaves the mathematical answer unchanged; scrambling individual coordinates creates another triangle.

    Nearly collinear points deserve care even if the denominator is not exactly zero. A very narrow triangle can produce a distant center, and small changes in measured coordinates can move that center substantially. Three-decimal output does not remove this sensitivity.

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Frequently Asked Questions

Is the circumcenter the same as the centroid?

Usually not. The centroid averages the three vertex coordinates and lies where the medians meet. The circumcenter is equally distant from the vertices. They coincide in an equilateral triangle, but that is a special case.

Why do three collinear points fail?

A finite circle cannot pass through three distinct points on a single straight line. The formula reflects this geometry through a zero denominator, so returning a single center would be misleading.

Can I change the order of the vertices?

Yes, as long as each complete x–y pair stays together. Relabeling A, B and C does not change the circle passing through the same three points.

How can I check a result without repeating the long formula?

Compute the distance from the returned center to each vertex. The three distances should agree within rounding. Use unrounded coordinates when available, because rounding the center can introduce small differences.

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Sources & References

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