What is the Equation of a Circle?

The equation of a circle is a mathematical way to describe all the points that lie on the circumference of a circle in a 2D Cartesian plane. Unlike a linear equation that forms a straight line, the circle equation is a quadratic relation in two variables, typically x and y.

Standard Form vs. General Form

There are two primary ways to represent a circle mathematically:

• Standard Form: (x - h)² + (y - k)² = r². Here, (h, k) represents the center of the circle, and r represents the radius. This form is highly useful because it allows you to instantly identify the circle's position and size.
• General Form: x² + y² + Dx + Ey + F = 0. This expanded form is often used in algebraic manipulation and calculus. Our calculator provides both formats along with the steps to convert between them.

How to Use the Circle Equation Calculator

Using this tool is straightforward. Simply follow these steps:

1. Enter the coordinates of the center (h, k). For example, if the center is at the origin, enter 0 for both.

2. Enter the radius of the circle. This must be a positive number.

3. Click "Generate Equation". The tool will automatically calculate the standard form, expand it into the general form, and show you exactly how the math was performed.

Frequently Asked Questions

What if I only have the diameter?
Simply divide the diameter by 2 to find the radius before entering it into the calculator.

Can the radius be zero?
If the radius is zero, the equation describes a single point (the center) rather than a circle.

How do I find the center from the general form?
You would need to use a process called "completing the square" for both the x and y terms to transform the general form back into the standard form.