Equation of a circle

Here you will learn about the equation of a circle, including how to recognize the equation of a circle, form an equation of a circle given its radius and center, use the equation of a circle to find its center and radius, and solve problems involving the equation of a circle.

Students will first learn about the equation of a circle as part of geometry in high school.

What is the equation of a circle?

The equation of a circle is katex is not defined where katex is not defined represents the radius, with a center at katex is not defined and katex is not defined where katex is not defined represents the radius, with a center at katex is not defined

The definition of a circle is a set of all points on a plane that are a fixed distance from a center. That distance is called the radius.

To understand the equation of a circle, consider the drawing below of a circle on a set of axes with the center at the origin.

Now consider a right angled triangle created when the radius of the circle is the hypotenuse of the right triangle.

• The horizontal line is the distance to the katex is not defined coordinate
• The vertical line is the distance to the katex is not defined coordinate
• The hypotenuse is the distance of the radius

You can now apply Pythagoras’ theorem to the above. Therefore, the general form of the equation of a circle centered at katex is not defined is:

katex is not defined

katex is not defined coordinate

katex is not defined coordinate

katex is not defined radius

For example,

Draw circle with equation katex is not defined

The circle has a center at katex is not defined

katex is not defined represents katex is not defined so the radius katex is not defined is katex is not defined

But what about a circle that is not centered on the origin? Let’s look at the same circle as above, but in a different position.

The radius of the circle below is still katex is not defined but now the center is at katex is not defined This means all the points on the circle have shifted up katex is not defined and right katex is not defined

Since katex is not defined is true for all circles a katex is not defined you can shift the circle back down to the origin.

To do that, you need to shift all the katex is not defined coordinates left katex is not defined on the katex is not definedaxis and shift all the katex is not defined coordinates down katex is not defined on the katex is not definedaxis to return this circle to the origin.

This leaves us with the equation katex is not defined

What about if this same circle was in another position?

The radius of the circle below is still katex is not defined but now the center is at katex is not defined This means all the points on the circle have shifted left katex is not defined and down katex is not defined

Since katex is not defined is true for all circles a katex is not defined you need to shift all the katex is not defined coordinates right katex is not defined and shift all the katex is not defined coordinates up katex is not defined to return this circle to the origin. This leaves us with the equation katex is not defined

Therefore, the general form of the equation of any circle is:

katex is not defined

katex is not defined coordinate of center of the circle

katex is not defined coordinate of center of the circle

katex is not defined radius of the circle

For example,

Draw the circle with equation katex is not defined

katex is not defined coordinate of center katex is not defined

katex is not defined coordinate of center katex is not defined

Coordinates of the center katex is not defined

katex is not defined length of radius katex is not defined

For example,

What is the equation of the circle with a center of katex is not defined and a radius of katex is not defined

The standard equation of a circle is katex is not defined

• The katex is not defined-coordinate of center of the circle is katex is not defined
• The katex is not defined-coordinate of the center of the circle is katex is not defined
• The radius is katex is not defined

So,

katex is not defined

What is the equation of a circle?

Common Core State Standards

How does this relate to high school math?

• Geometry – Expressing Geometric Properties with Equations (HSG.GPE.A.1)
  Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

How to use the equation of a circle

In order to use the equation of a circle:

1. Write the general equation of a circle.
2. State any variables you know.
3. Substitute any values you know into the equation.
4. Use the information you have to solve the problem.
5. Clearly state the answer.

Equation of a circle examples

Example 1: writing the equation of a circle with center at the origin

What is the equation of the circle with a center at the origin?

1. Write the general equation of a circle.

katex is not defined for circles at the origin.

2State any variables you know.

The distance from the center katex is not defined to the circle is katex is not defined

Radius katex is not defined

3Substitute any values you know into the equation.

katex is not defined

4Use the information you have to solve the problem.

Simplify the equation by squaring the radius.

katex is not defined

5Clearly state the answer.

The equation of the circle is: katex is not defined

Example 2: finding the radius of a circle given its equation

What is the radius of the circle with the equation katex is not defined

Example 3: writing the equation of a circle with center at the origin

What is the equation of a circle with a radius of katex is not defined and a center at the origin?

Example 4: given points – find the center of a circle

What is the equation of the circle with diameter endpoints at katex is not defined and katex is not defined

Example 5: finding the center and radius of a circle given its equation

What is the center and radius of the circle with the equation

katex is not defined

Example 6: state the equation given the center and radius

What is the equation of a circle with a radius of katex is not defined and a center at katex is not defined

Teaching tips for equation of a circle

• Be sure that students have a good understanding of coordinate geometry before asking them to graph circles and derive their equations or proofs.

• Let students use a graphing calculator to explore how changing the values for katex is not defined and katex is not defined change a circle.

Easy mistakes to make

• Forgetting the radius is squared in the equation
  Remember, in the equation the radius is shown as a squared value. To find the radius, you need to square root this value.

• Thinking the radius can be negative
  The radius cannot be negative because the radius is a length; it must always be a positive value.

• Confusing the location of the center with katex is not defined
  The equation indicates how a circle would need to shift to return to the origin.
  Therefore, the signs in the equation are opposite the signs in the coordinate.
  For example,
  An origin of katex is not defined would need to shift left katex is not defined and up katex is not defined which is
  katex is not defined

• Confusing equation of a circle with quadratic equations
  The equation of a circle and a quadratic equation are not the same. A quadratic equation appears on a graph as a parabola, not a circle.
  For example,
  katex is not defined is a quadratic equation.
  This equation can be factored as katex is not defined
  While they share some symbol commonalities, this is NOT the same as katex is not defined or katex is not defined

• Not recognizing the equation in other forms
  You may need to simplify, combine like terms or balance the sides of the equation to get the standard form of the equation of a circle.

Related types of graphs lessons

• Linear graph
• Cubic function graph
• Exponential function graph
• Reciprocal graph

The next lessons are

• Graphing linear equations
• Rate of change
• Systems of equations

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