Find Equation Of Circle Given 3 Points Calculator

Circle Equation Calculator

Enter the coordinates of three distinct, non-collinear points to find the equation of the circle passing through them.

What is a Find Equation of Circle Given 3 Points Calculator?

A find equation of circle given 3 points calculator is a tool used to determine the equation of a circle that passes through three given distinct points in a Cartesian coordinate system. If the three points are not collinear (do not lie on the same straight line), there is a unique circle that passes through them. This calculator finds both the standard form (x – h)² + (y – k)² = r² and the general form x² + y² + 2gx + 2fy + c = 0 of the circle’s equation, along with the center (h, k) and radius r.

This tool is useful for students learning analytical geometry, engineers, designers, and anyone needing to define a circle based on three specific points on its circumference. It automates the process of solving the system of equations or finding the intersection of perpendicular bisectors, which can be tedious to do manually.

Who should use it?

• Geometry students learning about circles.
• Engineers and architects in design and planning.
• Programmers working on graphics or geometric applications.
• Anyone needing to find a circle passing through three known locations.

Common Misconceptions

A common misconception is that any three points will define a circle. While three non-collinear points define a unique circle, if the three points lie on a straight line (are collinear), a circle cannot pass through all three. In such cases, the “circle” would have an infinite radius, degenerating into the line itself, or the denominator in the calculations becomes zero, indicating no finite center.

Find Equation of Circle Given 3 Points Calculator Formula and Mathematical Explanation

Given three non-collinear points P1(x1, y1), P2(x2, y2), and P3(x3, y3), we want to find the equation of the circle passing through them. The general equation of a circle is x² + y² + 2gx + 2fy + c = 0, where the center is (-g, -f) and the radius is √(g² + f² – c). Alternatively, the standard form is (x – h)² + (y – k)² = r², where the center is (h, k) and radius is r.

Since the three points lie on the circle, they satisfy its equation:

1. x1² + y1² + 2gx1 + 2fy1 + c = 0 => 2gx1 + 2fy1 + c = -(x1² + y1²)
2. x2² + y2² + 2gx2 + 2fy2 + c = 0 => 2gx2 + 2fy2 + c = -(x2² + y2²)
3. x3² + y3² + 2gx3 + 2fy3 + c = 0 => 2gx3 + 2fy3 + c = -(x3² + y3²)

This is a system of three linear equations in three variables (g, f, c). We can solve it using determinants (Cramer’s rule) or by finding the intersection of the perpendicular bisectors of the chords formed by the points.

Using the perpendicular bisector method:

• The perpendicular bisector of the chord P1P2 passes through the center.
• The perpendicular bisector of the chord P2P3 also passes through the center.
• The intersection of these two lines is the center (h, k) of the circle.
• The radius r is the distance from the center (h, k) to any of the three points, e.g., r = √((x1 – h)² + (y1 – k)²).

The system of equations can be solved for g, f, and c:
The determinant D = 4*x1*y2 – 4*x1*y3 – 4*y1*x2 + 4*y1*x3 + 4*x2*y3 – 4*x3*y2. If D is zero, the points are collinear.
If D is non-zero, g, f, and c can be found, then h = -g, k = -f, and r = √(h² + k² – c). Our find equation of circle given 3 points calculator uses these principles.

Variables Table

Variable	Meaning	Unit	Typical Range
(x1, y1)	Coordinates of the first point	None (or length units)	Real numbers
(x2, y2)	Coordinates of the second point	None (or length units)	Real numbers
(x3, y3)	Coordinates of the third point	None (or length units)	Real numbers
(h, k)	Coordinates of the circle’s center	None (or length units)	Real numbers
r	Radius of the circle	None (or length units)	Positive real numbers
g, f, c	Coefficients in the general equation x² + y² + 2gx + 2fy + c = 0	Varies	Real numbers

Variables used in the find equation of circle given 3 points calculator.

Practical Examples (Real-World Use Cases)

Example 1: Locating an Epicenter

Three seismic stations A(3, 4), B(-1, 2), and C(5, -2) detect an earthquake. They are roughly equidistant from the epicenter. We can use the find equation of circle given 3 points calculator to estimate the epicenter location (center of the circle) and the distance (radius).

Inputs: P1=(3, 4), P2=(-1, 2), P3=(5, -2)

Using the calculator, we would find the center (h, k) and radius r, giving the location and approximate distance to the epicenter.

Example 2: Designing a Circular Path

An architect wants to design a circular pathway that touches three specific points in a garden: P1(1, 0), P2(-1, 0), and P3(0, 1).

Inputs: P1=(1, 0), P2=(-1, 0), P3=(0, 1)

The find equation of circle given 3 points calculator gives:

• Center (h, k) = (0, 0)
• Radius r = 1
• Standard Equation: x² + y² = 1
• General Equation: x² + y² – 1 = 0

This helps in accurately plotting the circular path.

How to Use This Find Equation of Circle Given 3 Points Calculator

1. Enter Point 1 Coordinates: Input the x (x1) and y (y1) coordinates of the first point.
2. Enter Point 2 Coordinates: Input the x (x2) and y (y2) coordinates of the second point.
3. Enter Point 3 Coordinates: Input the x (x3) and y (y3) coordinates of the third point.
4. Calculate: The calculator automatically updates the results as you type. You can also click the “Calculate” button.
5. Read Results: The calculator will display:
   • The standard equation (x – h)² + (y – k)² = r²
   • The general equation x² + y² + 2gx + 2fy + c = 0
   • The coordinates of the center (h, k)
   • The radius r
   • The values of g, f, and c
   • A visual representation on the canvas
6. Collinearity Check: If the points are collinear or very close to it, an error message will be shown, as a unique circle cannot be defined.
7. Reset: Click “Reset” to clear the inputs to default values.
8. Copy Results: Click “Copy Results” to copy the main equations, center, and radius to your clipboard.

Key Factors That Affect Find Equation of Circle Given 3 Points Calculator Results

1. Collinearity of Points: If the three points lie on or very close to a straight line, it’s impossible to define a unique circle. The denominator in the calculations approaches zero, leading to very large or undefined values for the center and radius. Our find equation of circle given 3 points calculator checks for this.
2. Distinctness of Points: The three points must be distinct. If any two points are the same, you effectively have only two points, which are insufficient to define a unique circle (an infinite number of circles can pass through two points).
3. Precision of Coordinates: The accuracy of the input coordinates directly impacts the accuracy of the calculated center, radius, and equation. Small errors in input can lead to larger deviations, especially if the points are close together.
4. Scale of Coordinates: Very large or very small coordinate values might lead to numerical precision issues in the calculations, although modern calculators try to handle this.
5. Numerical Stability: The method used (solving linear equations or intersecting perpendicular bisectors) can have different numerical stability properties, especially with near-collinear points.
6. Computational Accuracy: The internal precision used by the calculator software or device can influence the final result, particularly for edge cases.

Frequently Asked Questions (FAQ)

Q1: What happens if the three points are collinear?
A1: If the three points lie on a straight line, a unique circle cannot be drawn through them. The find equation of circle given 3 points calculator will indicate that the points are collinear, and no standard circle equation will be provided because the radius would be infinite.

Q2: What if two of the points are the same?
A2: If two points coincide, you essentially have only two distinct points, and an infinite number of circles can pass through two points. The calculator might produce an error or an indeterminate result.

Q3: Does the order of the points matter?
A3: No, the order in which you enter the three points does not affect the final equation of the circle.

Q4: Can I use the calculator for points with decimal coordinates?
A4: Yes, the find equation of circle given 3 points calculator accepts decimal values for the coordinates.

Q5: What is the difference between the standard and general form of the circle equation?
A5: The standard form is (x – h)² + (y – k)² = r², which directly shows the center (h, k) and radius r. The general form is x² + y² + 2gx + 2fy + c = 0, where g = -h, f = -k, and c = h² + k² – r². The general form is useful for certain algebraic manipulations.

Q6: How does the calculator find the center and radius?
A6: It typically solves a system of three linear equations derived from the general circle equation by substituting the coordinates of the three points, or it finds the intersection point of the perpendicular bisectors of the chords connecting the points.

Q7: Can I use this calculator for 3D points?
A7: No, this find equation of circle given 3 points calculator is specifically for points in a 2D Cartesian plane (x, y coordinates). Finding a circle or sphere through points in 3D requires different methods.

Q8: What if the points form a very small or very large circle?
A8: The calculator should handle a wide range of coordinate values, but extremely large or small numbers might test the limits of floating-point precision, potentially affecting the accuracy of the displayed results.

Related Tools and Internal Resources

• Distance Between Two Points Calculator: Calculate the distance between any two points in a 2D plane.
• Midpoint Calculator: Find the midpoint between two points.
• Slope Calculator: Determine the slope of a line connecting two points.
• Equation of a Line Calculator: Find the equation of a line given two points or other properties.
• Circle Area Calculator: Calculate the area of a circle given its radius.
• Circle Circumference Calculator: Calculate the circumference of a circle.