What Is the Equation of a Circle?
What Is the Equation of a Circle?
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.
The image below shows what we mean by a point on a circle centered at the origin and its radius:
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.