The LessonThe equation of a circle (centered on the origin) is in the form:
In this equation,
- x and y are the Cartesian coordinates of points on the (boundary of the) circle.
- r is the radius of the circle.
Real Examples of Equations of CirclesIt is easier to understand the equation of a circle with examples.
A circle with a radius of 4 will have the equation:
A circle with a radius of 2 will have the equation:
A circle with a radius of 9 will have the equation:
Understanding the Equation of a CircleA circle is a set of points. Each point can be described using Cartesian coordinates (x, y). The equation of a circle x2 + y2 = r2 is true for all points on the circle. It gives the relationship between the x-coordinate and y-coordinate of each point on the circle and the radius of the circle. Consider a circle with a radius of 2. Its equation is:
x2 + y2 = 4Let us consider some points on the circle.
(2, 0)Consider the point at (2, 0). It has a x-coordinate of 2 and a y-coordinate of 0.
At this point x = 2 and y = 0. Inserting these values into the equation:
22 + 02 = 4The equation is satisfied ✔.
(√2, √2)Consider the point at (√2, √2). It has a x-coordinate of √2 and a y-coordinate of √2.
At this point x = √2 and y = √2. Inserting these values into the equation:
√22 + √22 = 2 + 2 = 4Again, the equation is satisfied ✔. Any point on the circle would satisfy the equation.
Lesson SlidesThe slider below explains why the "Equation of a Circle Works". Open the slider in a new tab
The Circle Must Be Centered at the OriginFor this equation to work, the circle must be centered at the origin of the graph:
The equation will not work if the circle is not centered at the origin of the graph:
Read more about how to find the equation of a circle not centered at the origin
Getting the Equation RightThe equation of a circle must have an x2 term and a y2 added together. These is not the equations of a circle:
Don't be fooled if the equation is simply rearranged. Below are equations of circle that can put into the familiar form with a little algebra: