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Equations of Circles

(x - h)^2 + (y-k)^2 = r^2

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Equations of Circles

What type of graph is produced by the equation ? How can you turn this equation into graphing form in order to graph it?

Equations of Circles 

A circle is one example of a conic section. A circle is the set of all points that are equidistant from a center point. The general equation of a circle is  where  is the center of the circle and  is the radius of the circle.

You can derive the equation of a circle with the help of the Pythagorean Theorem.

 

Deriving the Equation of a Circle  

1. Label the three sides of the right triangle below in terms of , and . Then, use the Pythagorean Theorem to write an equation that shows the relationship between the three sides.

The three sides of the triangle are , , and , where  is the hypotenuse.

By the Pythagorean Theorem, .

2. Do all points on the circle from the previous problem satisfy the equation ?

Yes, anywhere you move the point , the lengths of the sides of the triangle will still be , and , so  will still be true.

In fact, the set of points that make up the circle is exactly the set of points that satisfy the equation . This is why the general equation of a circle centered at the origin is , where  is the radius of the circle.

3. Why does a circle with center  and radius  have the general equation ?

Pick a point on the circle and draw a right triangle connecting that point with the center of the circle.

The three sides of the triangle have lengths and .

The relationship between the sides of the triangle is shown with the Pythagorean Theorem:

Points on the other side of the circle produce triangles with slightly different side lengths:

However, the relationship between the three sides of this triangle is the same, as shown below:

The set of points that satisfy the equation  are the set of points that make up the circle.

Examples

Example 1

Earlier, you were asked how you can turn  into graphing form in order to graph it. 

The equation  is the equation for a circle. You can tell it is a circle because there is both an  term and a  term, and the coefficients of each are the same. To turn this equation into the form , you must complete the square twice. Remember that when completing the square, you are trying to figure out what number to add to make a perfect square trinomial. In order to maintain the equality of the equation, you must add the same numbers to both sides of the equation.

Example 2

What is the equation of a circle centered at  with a radius of ?

, , . The equation of the circle is . This can be simplified to . Notice that the signs of the 4 and the 5 in the equation are opposite from the signs of the 4 and the 5 in the center point.

Example 3

What are the center and radius of the circle described by the equation:

Complete the square to rewrite the equation in graphing form:

The center of the circle is  and the radius is 4.

Example 4

Graph the circle from #3.

 

Review

Graph the following circles:

1.

2.

3.

4.

5.

Find the center and radius of the circle described by each equation.

6.

7.

8.

9.

10.

11. Explain why a circle with radius  that is centered at the origin will have the equation .

12. Use the Pythagorean Theorem to help explain why a circle with a center in the second quadrant with radius  and center  will have the equation . Note that in the second quadrant the value for  will be negative and the value for  will be positive.

13. Use the Pythagorean Theorem to help explain why a circle with a center in the third quadrant with radius  and center  will have the equation . Note that in the third quadrant the value for  will be negative and the value for  will be negative.

14. Write the equation of the following circle:

15. Write the equation of the following circle:

 

Review (Answers)

To see the Review answers, open this PDF file and look for section 10.2. 

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Vocabulary

center

The center of a circle is the point that defines the location of the circle. All points on the circle are equidistant from the center of the circle.

Circle

A circle is the set of all points at a specific distance from a given point in two dimensions.

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by a^2 + b^2 = c^2, where a and b are legs of the triangle and c is the hypotenuse of the triangle.

Radius

The radius of a circle is the distance from the center of the circle to the edge of the circle.

Vertical Line Test

The vertical line test says that if a vertical line drawn anywhere through the graph of a relation intersects the relation in more than one location, then the relation is not a function.

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