Finding Radius And Center Calculator

Enter the coordinates of three distinct, non-collinear points to find the center (h, k) and radius (r) of the circle passing through them.

Point	X-coordinate	Y-coordinate
Point 1	1	7
Point 2	8	6
Point 3	7	-1

Input coordinates used for the calculation.

What is a Circle From Three Points Calculator?

A circle from three points calculator is a tool used to determine the unique circle that passes through three given, non-collinear points in a 2D Cartesian plane. It calculates the coordinates of the center (h, k) and the radius (r) of this circle. This is also known as finding the circumcircle of the triangle formed by the three points, with the center being the circumcenter.

Anyone working with geometry, computer graphics, engineering, or physics might need to find the circle defined by three points. For example, it's used in fitting circular arcs to data points, determining the path of objects moving in a circle, or in certain geometric constructions. This circle from three points calculator simplifies the process.

A common misconception is that any three points define a circle. While three non-collinear points define a unique circle, if the three points lie on a straight line (are collinear), an infinite number of circles or no circle of finite radius can pass through them in a standard sense (or rather, a line can be considered a circle of infinite radius). Our circle from three points calculator handles cases where points might be nearly collinear.

Circle From Three Points Formula and Mathematical Explanation

The general equation of a circle is (x – h)² + (y – k)² = r², where (h, k) is the center and r is the radius. Expanding this, we get x² – 2hx + h² + y² – 2ky + k² = r², or x² + y² – 2hx – 2ky + (h² + k² – r²) = 0.

We can rewrite this as x² + y² + 2gx + 2fy + c = 0, where h = -g, k = -f, and r² = g² + f² – c.

Given three points (x₁, y₁), (x₂, y₂), and (x₃, y₃), each point must satisfy the circle 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 g, f, and c. We can solve it using various methods, including determinants (Cramer's rule) or by finding the intersection of the perpendicular bisectors of the chords formed by the points.

Using perpendicular bisectors: The center of the circle is equidistant from all three points, so it lies on the perpendicular bisector of each chord connecting two points. We find the equations of two such perpendicular bisectors and their intersection point gives the center (h, k). The radius is then the distance from the center to any of the three points.

For example, the perpendicular bisector of the line segment joining (x₁, y₁) and (x₂, y₂) has the equation derived from (x-x₁)²+(y-y₁)² = (x-x₂)²+(y-y₂)².

The circle from three points calculator above uses the determinant method derived from the linear equations for g, f, and c.

Variables Table

Variable	Meaning	Unit	Typical Range
(x₁, y₁), (x₂, y₂), (x₃, y₃)	Coordinates of the three points	Length units	Any real numbers
(h, k)	Coordinates of the center of the circle	Length units	Any real numbers
r	Radius of the circle	Length units	r > 0
g, f, c	Coefficients in x² + y² + 2gx + 2fy + c = 0	Varies	Any real numbers

Practical Examples (Real-World Use Cases)

Let's see how the circle from three points calculator works with examples.

Example 1: Archeological Site

An archaeologist finds three artifacts at coordinates (1, 7), (8, 6), and (7, -1) relative to a reference point. They suspect these artifacts were placed on the perimeter of a circular structure.

• Point 1: (1, 7)
• Point 2: (8, 6)
• Point 3: (7, -1)

Using the circle from three points calculator with these inputs, we find:

• Center (h, k) ≈ (4, 3)
• Radius (r) ≈ 5
• Equation: (x – 4)² + (y – 3)² = 25

This suggests the circular structure had its center near (4, 3) and a radius of about 5 units.

Example 2: Satellite Dish Design

An engineer is designing a parabolic dish, but first needs to fit a circular rim through three points: (-5, 0), (5, 0), and (0, 1.25) to ensure it's wide enough and has the correct initial curvature.

• Point 1: (-5, 0)
• Point 2: (5, 0)
• Point 3: (0, 1.25)

The circle from three points calculator gives:

• Center (h, k) ≈ (0, -9.375)
• Radius (r) ≈ 10.625
• Equation: x² + (y + 9.375)² ≈ 112.89

The center is on the y-axis, as expected due to symmetry, and the radius is quite large, indicating a shallow curve through these points.

How to Use This Circle From Three Points Calculator

1. Enter Point Coordinates: Input the x and y coordinates for each of the three points (x1, y1), (x2, y2), and (x3, y3) into the designated fields.
2. Calculate: Click the "Calculate" button or simply change any input value. The calculator will automatically compute the results if the inputs are valid.
3. View Results: The calculator will display:
   • The coordinates of the center (h, k).
   • The radius (r) of the circle.
   • The equation of the circle.
   • Intermediate values like g, f, c if needed for verification.
   • A visualization of the points and the circle.
4. Check for Collinearity: If the points are collinear or very close to it, the calculator will indicate that a unique circle cannot be determined (or it has an infinitely large radius).
5. Reset: Use the "Reset" button to clear the inputs to their default values.
6. Copy Results: Use the "Copy Results" button to copy the main outputs to your clipboard.

When reading the results, pay attention to the center and radius. These define the circle completely. The equation is just another way to represent the same circle.

Key Factors That Affect Circle From Three Points Results

1. Coordinates of the Points: The most direct factor. Changing any coordinate will change the circle's center and radius.
2. Collinearity of Points: If the three points lie on or very close to a straight line, the radius will become very large, and the center will be very far away. Mathematically, a unique circle of finite radius cannot be drawn through three collinear points. The calculator will warn about this.
3. Distance Between Points: If the points are very close together, small errors in their coordinates can lead to large changes in the calculated center and radius, indicating sensitivity.
4. Symmetry of Points: If the points have some symmetry (e.g., forming an isosceles or equilateral triangle), the center of the circle will also exhibit related symmetry. For example, if two points have the same y-value, the center will lie on the perpendicular bisector which is a vertical line midway between them.
5. Numerical Precision: The calculations involve floating-point arithmetic. Very large or very small coordinate values, or near-collinearity, might test the limits of precision, though our circle from three points calculator aims for high accuracy.
6. Distinctness of Points: The three points must be distinct. If any two points are the same, they don't provide enough information to define a unique circle (an infinite number of circles can pass through two points).

Frequently Asked Questions (FAQ)

Q1: What happens if the three points are collinear? A1: If the three points lie on a straight line, they do not define a unique circle with a finite radius. The "circle" would either be the line itself (infinite radius) or it's impossible to draw one through them in the standard sense. Our circle from three points calculator will detect this and inform you.

Q2: Can I use negative coordinates? A2: Yes, the x and y coordinates of the points can be positive, negative, or zero.

Q3: How accurate is this circle from three points calculator? A3: The calculator uses standard floating-point arithmetic, providing good accuracy for most practical purposes. However, with nearly collinear points or extremely large/small coordinates, precision limitations might become noticeable.

Q4: What if two of my points are the same? A4: If two points coincide, you effectively only have two distinct points, and an infinite number of circles can pass through them. The calculator may give an error or an indeterminate result. Ensure you input three *distinct* points.

Q5: What is the circumcircle? A5: The circle passing through the three vertices of a triangle is called its circumcircle. Our circle from three points calculator finds this circumcircle if you consider the three points as the vertices of a triangle.

Q6: Where is the center of the circle located relative to the triangle formed by the points? A6: The center of the circle (the circumcenter) is the intersection of the perpendicular bisectors of the sides of the triangle formed by the three points. It can be inside, on, or outside the triangle, depending on whether the triangle is acute, right-angled, or obtuse, respectively.

Q7: What is the formula used by the circle from three points calculator? A7: It solves the system of linear equations derived by substituting the coordinates of the three points into the general circle equation x² + y² + 2gx + 2fy + c = 0 to find g, f, and c, and then h=-g, k=-f, r=sqrt(g²+f²-c). Alternatively, it finds the intersection of the perpendicular bisectors.

Q8: Can this calculator handle very large or very small coordinate values? A8: Yes, within the limits of standard JavaScript number representation. If coordinates are extremely large or small, you might encounter precision issues, especially if the points are also nearly collinear.

Related Tools and Internal Resources

• Distance Calculator: Calculate the distance between two points, useful for finding the radius once the center is known.
• Midpoint Calculator: Find the midpoint of a line segment, used in finding perpendicular bisectors.
• Circle Equation Explorer: Learn more about the different forms of the circle equation.
• Geometry Formulas: A collection of useful geometry formulas, including those for circles and triangles.
• Area Calculator: Calculate the area of various shapes, including circles.
• Volume Calculator: For 3D shapes related to circles, like cylinders and spheres.