Equation of a Circle
(KS2, Year 6)

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The equation of a circle (centered on the origin) is in the form:

basic circle equation

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.

The image below shows what we mean by a point on a circle centered at the origin and its radius:

basic circle equation image

Real Examples of Equations of Circles

It 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 Circle

A 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 = 4

Let 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.

circle 2 0

At this point x = 2 and y = 0. Inserting these values into the equation:

2² + 0² = 4

The equation is satisfied ✔.

(√2, √2)

Consider the point at (√2, √2). It has a x-coordinate of √2 and a y-coordinate of √2.

circle root 2 root 2

At this point x = √2 and y = √2. Inserting these values into the equation:

√2² + √2² = 2 + 2 = 4

Again, the equation is satisfied ✔. Any point on the circle would satisfy the equation.

Lesson Slides

The slider below explains why the "Equation of a Circle Works".

Beware

The Circle Must Be Centered at the Origin

For this equation to work, the circle must be centered at the origin of the graph:

circle centered at origin

The equation will not work if the circle is not centered at the origin of the graph:

circle not centered at origin

how to find the equation of a circle not centered at the origin

Getting the Equation Right

The equation of a circle must have an x² term and a y² added together. These is not the equations of a circle:

not circle equations

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:

rearranged circle equations