Find Equation of Circle Calculator
This calculator helps you find the equation of a circle. You can input the center and radius, or three points on the circle's circumference. Our Find Equation of Circle Calculator is easy to use.
What is a Find Equation of Circle Calculator?
A Find Equation of Circle Calculator is a tool designed to determine the standard and general form equations of a circle based on certain given parameters. You can usually find the equation of a circle if you know its center coordinates (h, k) and its radius (r), or if you know the coordinates of three distinct points that lie on the circle's circumference. The calculator automates the mathematical steps involved, providing you with the equation quickly and accurately. This Find Equation of Circle Calculator is useful for students, engineers, and anyone dealing with geometric problems involving circles.
The standard form equation is (x – h)² + (y – k)² = r², while the general form is x² + y² + 2gx + 2fy + c = 0. Our Find Equation of Circle Calculator gives both.
Who should use it?
Students studying geometry or algebra, mathematicians, engineers, architects, and designers often use a Find Equation of Circle Calculator to solve problems, verify calculations, or design systems involving circular components.
Common misconceptions
A common misconception is that any three points define a unique circle. While this is true for non-collinear points, if the three points lie on a straight line, they do not define a circle (or they define a circle with infinite radius, which is a line). Another is confusing the radius with the diameter or the standard form with the general form. The Find Equation of Circle Calculator helps clarify these.
Find Equation of Circle Calculator Formula and Mathematical Explanation
There are two primary ways to define the equation of a circle, leading to two forms:
1. Using Center (h, k) and Radius (r):
The standard form (or center-radius form) of the equation of a circle with center (h, k) and radius r is:
(x – h)² + (y – k)² = r²
This equation represents the definition of a circle: the set of all points (x, y) that are at a fixed distance r (the radius) from a fixed point (h, k) (the center).
Expanding this, we get:
x² – 2hx + h² + y² – 2ky + k² = r²
x² + y² – 2hx – 2ky + (h² + k² – r²) = 0
This leads to the general form: x² + y² + 2gx + 2fy + c = 0, where g = -h, f = -k, and c = h² + k² – r².
2. Using Three Non-Collinear Points (x1, y1), (x2, y2), (x3, y3):
If we have three points, we can use the general form: x² + y² + 2gx + 2fy + c = 0. Substituting the coordinates of the three points, we get a system of three linear equations in g, f, and c:
2gx₁ + 2fy₁ + c = -(x₁² + y₁²)
2gx₂ + 2fy₂ + c = -(x₂² + y₂²)
2gx₃ + 2fy₃ + c = -(x₃² + y₃²)
We solve this system for g, f, and c. Then, the center is (-g, -f) and the radius is r = √(g² + f² – c). If g² + f² – c < 0, the points do not form a real circle. If the points are collinear, the determinant of the system's matrix is zero, and no unique circle passes through them.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| h | x-coordinate of the center | Length units | Any real number |
| k | y-coordinate of the center | Length units | Any real number |
| r | Radius of the circle | Length units | Positive real number (r > 0) |
| g | Coefficient in general form (-h) | Length units | Any real number |
| f | Coefficient in general form (-k) | Length units | Any real number |
| c | Constant in general form (h² + k² – r²) | Length units squared | Any real number |
| x₁, y₁ … x₃, y₃ | Coordinates of points on the circle | Length units | Any real number |
Our distance formula calculator can be useful here.
Practical Examples (Real-World Use Cases)
Example 1: Center and Radius Given
Suppose an engineer is designing a circular gear with a center at (2, -3) and a radius of 5 units.
- h = 2, k = -3, r = 5
- Standard Equation: (x – 2)² + (y – (-3))² = 5² => (x – 2)² + (y + 3)² = 25
- General Equation: x² – 4x + 4 + y² + 6y + 9 = 25 => x² + y² – 4x + 6y – 12 = 0
The Find Equation of Circle Calculator would output both forms.
Example 2: Three Points Given
Imagine locating an epicenter based on seismic readings from three stations at points A(1, 2), B(3, 4), and C(-1, 4). Assuming the epicenter is equidistant from these stations, it lies at the center of a circle passing through them.
- (x1, y1) = (1, 2), (x2, y2) = (3, 4), (x3, y3) = (-1, 4)
- Using the Find Equation of Circle Calculator with these points, we'd solve the system of equations to find g, f, c, then h, k, r.
- For these points, the center (h, k) is (1, 4) and radius r is 2.
- Standard Equation: (x – 1)² + (y – 4)² = 4
- General Equation: x² + y² – 2x – 8y + 13 = 0
The Find Equation of Circle Calculator provides these equations.
How to Use This Find Equation of Circle Calculator
- Select Mode: Choose whether you want to input the "Center & Radius" or "Three Points".
- Enter Values:
- If "Center & Radius": Enter the x-coordinate (h), y-coordinate (k) of the center, and the radius (r).
- If "Three Points": Enter the x and y coordinates for each of the three points (x1, y1), (x2, y2), (x3, y3).
- Calculate: Click the "Calculate" button or observe the results updating as you type.
- Read Results: The calculator will display:
- The standard equation of the circle.
- The general equation of the circle.
- The calculated center (h, k) and radius (r) (especially useful in 3-point mode).
- The general form coefficients g, f, and c.
- Visualize: The canvas will show a graphical representation of the circle.
- Reset: Click "Reset" to clear inputs and results to default values.
- Copy: Click "Copy Results" to copy the equations and key values.
The Find Equation of Circle Calculator makes it simple to get the circle's equation.
Key Factors That Affect Find Equation of Circle Calculator Results
- Center Coordinates (h, k): The position of the center directly shifts the circle on the coordinate plane, affecting the 'h' and 'k' values in the standard equation and 'g' and 'f' in the general form.
- Radius (r): The radius determines the size of the circle and appears as r² in the standard equation and influences 'c' in the general form. A larger radius means a larger circle.
- Coordinates of the Three Points: If using the three-point method, the precise location of these points determines the center and radius. If the points are collinear, a circle cannot be uniquely determined (or it's a line). The Find Equation of Circle Calculator will indicate this.
- Accuracy of Input: Small errors in the input coordinates or radius can lead to different equations. Ensure your inputs are accurate.
- Non-Collinearity of Points: For the three-point method, the points must not lie on the same straight line for a unique circle to be defined. The Find Equation of Circle Calculator checks for this.
- Real vs. Imaginary Radius: When solving from three points, if g² + f² – c < 0, the radius would be imaginary, meaning no real circle passes through those points under the standard definition.
Understanding these factors helps in interpreting the results from the Find Equation of Circle Calculator and relating them to the circle area calculator or circumference calculator.
Frequently Asked Questions (FAQ)
- What is the standard equation of a circle?
- The standard equation of a circle with center (h, k) and radius r is (x – h)² + (y – k)² = r².
- What is the general form of the equation of a circle?
- The general form is x² + y² + 2gx + 2fy + c = 0, where h=-g, k=-f, and r²=g²+f²-c.
- How do you find the equation of a circle given three points?
- Substitute the coordinates of the three points into the general form equation to get three linear equations in g, f, and c. Solve this system to find g, f, and c, then h, k, and r. Our Find Equation of Circle Calculator automates this.
- Can any three points define a circle?
- Only if the three points are not collinear (do not lie on the same straight line). If they are collinear, they define a line, not a unique circle.
- What if the radius calculated from three points is zero or imaginary?
- If r=0, the "circle" is a single point. If r² < 0, there is no real circle passing through the given points.
- How does this Find Equation of Circle Calculator handle collinear points?
- It checks for collinearity and will display an error message if the three input points lie on a line, as a unique circle cannot be formed.
- Can I use negative coordinates for the center or points?
- Yes, the coordinates (h, k) and (x1, y1), etc., can be positive, negative, or zero.
- What is the relationship between the center-radius form and the general form?
- The general form is derived by expanding the center-radius form. You can convert between them using g=-h, f=-k, c=h²+k²-r².
You might also find our midpoint calculator useful.
Related Tools and Internal Resources
- Circle Area Calculator: Calculates the area of a circle given its radius.
- Circumference Calculator: Calculates the circumference of a circle given its radius.
- Midpoint Calculator: Finds the midpoint between two points.
- Distance Formula Calculator: Calculates the distance between two points in a plane.
- Slope Calculator: Determines the slope of a line given two points.
- Pythagorean Theorem Calculator: Solves for sides of a right triangle.
These tools, including the Find Equation of Circle Calculator, can help with various geometry problems.