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What Is the Basic Equation of a Circle? (Mathematics Lesson)

What Is the Basic Equation of a Circle?

The equation of a circle centered at the origin is:



where r is the radius of the circle.

Real Examples of Equations of Circles

  • 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:


Why the Equation of a Circle Works

The slider below explains the equation of a circle:

The Equation of a Circle that Is Not Centered at the Origin

The equation of a circle discussed above is for circles centered at the origin.

If a circle is not centered at the origin, but at a point (a, b), the equation of a circle is:





Read more about the equation of a circle that is not centered at the origin

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Note
WHAT IS A CIRCLE?

A circle is a shape containing a set of points that are all the same distance from a given point, its center.



WHAT DOES AN EQUATION OF A CIRCLE MEAN?

A circle is a set of points.

Each point can be described using Cartesian coordinates: (x, y).



For each point there is a relationship between x and y, given by the equation:



This is true for all points on the circle.

For example, a circle has a radius of 2. Its equation is:

x2 + y2 = 4

Consider the point at (2, 0).



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

22 + 02 = 4

The equation is satisfied.

Consider another point (√2, √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.

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:



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: