Equation of a Circle Calculator: Find Center & Radius

This calculator works out the standard equation of a circle from three points on its circumference, or from its center and radius. You enter either three (x, y) coordinate pairs or a center point and a radius, and you get the equation in the form (x − h)² + (y − k)² = r², along with the center coordinates and the radius value. If you only have the equation itself, it also reverses the process to extract the center and radius for you.

Formula

The standard equation of a circle with center (h, k) and radius r is:

(x − h)² + (y − k)² = r²

Here, x and y are the coordinates of any point on the circle, h and k are the coordinates of the center, and r is the radius (all in the same unit, e.g., meters, feet, or just plain coordinate units). If the center is at the origin (0, 0), the equation simplifies to x² + y² = r².

When you know three points on the circle, the calculator solves for h and k using the perpendicular bisectors of the chords. The formulas used are:

h = [(x�² + y�²)(y₂ − y₃) + (x₂² + y₂²)(y₃ − y�) + (x₃² + y₃²)(y� − y₂)] / [2(x�(y₂ − y₃) + x₂(y₃ − y�) + x₃(y� − y₂))]

k = [(x�² + y�²)(x₃ − x₂) + (x₂² + y₂²)(x� − x₃) + (x₃² + y₃²)(x₂ − x�)] / [2(x�(y₂ − y₃) + x₂(y₃ − y�) + x₃(y� − y₂))]

Once h and k are known, the radius is simply the distance from the center to any of the three points: r = √((x� − h)² + (y� − k)²).

Worked Example

Using the default values from the calculator: Point 1 = (1, 2), Point 2 = (5, 6), Point 3 = (3, 8).

1. First, compute the denominator D = 2[x�(y₂ − y₃) + x₂(y₃ − y�) + x₃(y� − y₂)] = 2[1(6−8) + 5(8−2) + 3(2−6)] = 2[1(−2) + 5(6) + 3(−4)] = 2[−2 + 30 − 12] = 2[16] = 32.
2. Now h = [(1²+2²)(6−8) + (5²+6²)(8−2) + (3²+8²)(2−6)] / 32 = [(5)(−2) + (61)(6) + (73)(−4)] / 32 = [−10 + 366 − 292] / 32 = 64 / 32 = 2.
3. For k, use the numerator: [(1²+2²)(3−5) + (5²+6²)(1−3) + (3²+8²)(5−1)] / 32 = [(5)(−2) + (61)(−2) + (73)(4)] / 32 = [−10 − 122 + 292] / 32 = 160 / 32 = 5.
4. Radius r = √((1−2)² + (2−5)²) = √(1 + 9) = √10 ≈ 3.16.

So the circle through (1,2), (5,6), and (3,8) has center (2, 5) and radius √10, giving the equation (x − 2)² + (y − 5)² = 10. The calculator shows r = 3.16 when rounded to two decimals.

How the Three-Point Method Works

Any three non-collinear points define a unique circle. The geometric idea is that the perpendicular bisector of any chord passes through the circle’s center. If you take the segment between point 1 and point 2, its perpendicular bisector is a line; the same for the segment between point 2 and point 3. The intersection of those two lines is the center. The algebraic formulas above are just the result of solving those two linear equations simultaneously. This is why the calculator asks for three points rather than two – two points only give a chord, and infinitely many circles can pass through them.

If you already have the center and radius, you don’t need the three-point method. Just plug them into the standard form. The calculator’s first output (h) is the x-coordinate of the center, the second (k) is the y-coordinate, and the third (r) is the radius. The headline answer is the center’s x-coordinate, but you can read all three together to write the full equation.

When to Use This Calculator

You’ll typically need this in coordinate geometry, computer graphics, or physics when you have three sample points from a circular path or object. For example:

• Finding the center of a circular arc from three measured points on a machine part.
• Determining the orbit circle from three positions of a satellite in a 2D plane.
• Checking whether a fourth point lies on the same circle by plugging it into the resulting equation.
• Converting a circle equation from expanded form (x² + y² + Dx + Ey + F = 0) back to standard form to read off the center and radius.

If you have the expanded form, you can also use the formulas h = −D/2 and k = −E/2, then r = √(h² + k² − F). The calculator here does not take the expanded coefficients directly, but you can compute h and k from them mentally or with a separate step, then enter those as center coordinates.

Common Mistakes and Pitfalls

• Mixing up x and y: Swapping the coordinates of a point changes the result completely. Double-check that you enter each pair in the correct order.
• Forgetting the sign in the equation: The standard form has minus signs: (x − h)². If your center is (−3, 4), the equation is (x + 3)² + (y − 4)² = r². A common error is writing (x − 3)² instead.
• Using diameter instead of radius: If a problem gives you the diameter, halve it first. The radius is half the diameter, and the equation uses r², so a wrong radius changes the right-hand side dramatically.
• Rounding too early: If you round h and k to two decimals before computing r, you may get a slightly different radius. The calculator keeps full precision internally and only rounds the displayed values.
• Collinear points: If the three points lie on a straight line, the denominator in the formula becomes zero, and no circle exists. The calculator will show an error or an undefined result. Always check that your points are not collinear.

Limits and Accuracy

The three-point method is exact for ideal mathematical points. In real-world measurements, small errors in the coordinates can shift the calculated center significantly, especially if the three points are close together or nearly collinear. The more spread out the points are, the more stable the result. If your points come from physical measurements, expect some noise; the calculator gives the exact circle that passes through the entered coordinates, not a best-fit circle. For noisy data, a least-squares fit would be more appropriate, but that is a different calculation. Also note that the formula assumes a standard Euclidean plane; it does not handle circles on a sphere (like the Earth’s surface) or in higher dimensions.

Frequently Asked Questions

What is the general equation of a circle?

The general equation is x² + y² + Dx + Ey + F = 0, where D, E, and F are constants. You can convert it to standard form by completing the square. The center is (−D/2, −E/2) and the radius is √((D/2)² + (E/2)² − F). This form is useful when you have an expanded polynomial rather than points.

Can I use this calculator if I only have the center and radius?

Yes, but you don’t need the three-point inputs. Simply enter the center coordinates as three points? No – the calculator is designed for three points. For center and radius, you can manually write the equation: (x − h)² + (y − k)² = r². If you want to verify, you can pick any two points on the circle and enter them along with the center as a third point, but that’s overly complicated. A simpler approach is to use the expanded form conversion mentioned above.

What if my three points are collinear?

If the points lie on a straight line, no circle passes through all three. The denominator in the formula becomes zero, and the calculator will not produce a valid result. Check your points for collinearity by seeing if the slopes between consecutive points are equal.

How do I find the radius if I have the equation in expanded form?

From x² + y² + Dx + Ey + F = 0, compute h = −D/2, k = −E/2, then r = √(h² + k² − F). For example, x² + y² − 4x − 10y + 20 = 0 gives h = 2, k = 5, and r = √(4 + 25 − 20) = √9 = 3. This matches the center and radius from the worked example if you scale appropriately.

Does the calculator work with negative coordinates?

Yes, you can enter negative x or y values. The formulas handle them correctly. Just make sure to include the minus sign when typing. The default minimum is 0, but you can type negative numbers anyway; the min only affects the spinner arrows, not the input itself.

What unit is the radius in?

The radius is in the same unit as the coordinates. If your points are in meters, the radius is in meters; if they are in feet, the radius is in feet. The calculator treats all numbers as dimensionless, so you must keep your units consistent across all three points.

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in x1 | Point 1 x | - | 1 | 0
in y1 | Point 1 y | - | 2 | 0
in x2 | Point 2 x | - | 5 | 0
in y2 | Point 2 y | - | 6 | 0
in x3 | Point 3 x | - | 3 | 0
in y3 | Point 3 y | - | 8 | 0
out h | Center x (h) | - | ((x1^2+y1^2)*(y2-y3)+(x2^2+y2^2)*(y3-y1)+(x3^2+y3^2)*(y1-y2))/(2*(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2))) | 2
out k | Center y (k) | - | ((x1^2+y1^2)*(x3-x2)+(x2^2+y2^2)*(x1-x3)+(x3^2+y3^2)*(x2-x1))/(2*(x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2))) | 2
out r | Radius | - | sqrt((x1-h)^2+(y1-k)^2) | 2