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Equation of a Circle
(KS2, Year 6)
The Lesson
The 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 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 x^{2} + y^{2} = r^{2} is true for all points on the circle. It gives the relationship between the xcoordinate and ycoordinate of each point on the circle and the radius of the circle. Consider a circle with a radius of 2. Its equation is:
x^{2} + y^{2} = 4
Let us consider some points on the circle.
(2, 0)
Consider the point at (2, 0). It has a xcoordinate of 2 and a ycoordinate of 0.At this point x = 2 and y = 0. Inserting these values into the equation:
2^{2} + 0^{2} = 4
The equation is satisfied ✔.
(√2, √2)
Consider the point at (√2, √2). It has a xcoordinate of √2 and a ycoordinate of √2.At this point x = √2 and y = √2. Inserting these values into the equation:
√2^{2} + √2^{2} = 2 + 2 = 4
Again, the equation is satisfied ✔.
Any point on the circle would satisfy the equation.
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: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 Right
The equation of a circle must have an x^{2} term and a y^{2} 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:
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