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,

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 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.

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.

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.

Why the Equation of a Circle Works

The slider below explains the equation of a circle.

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See Also

What is an equation? What is a circle? What are Cartesian coordinates? What is the x-coordinate? What is the y-coordinate?