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Math Lesson 15.6.2 - Equation of a Circle

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Welcome to our Math lesson on Equation of a Circle, this is the second lesson of our suite of math lessons covering the topic of Circle Graphs, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Equation of a Circle

In tutorial 14.5, we explained how to calculate the distance between two points A and B. The formula used for this was

dAB = √(xB - xA )2 + (yB - yA )2

Since the radii of a circle are all equal, the distance from the centre to any point of the circle is constant. Therefore, we can use the same approach as when calculating the distance between two points. In this case, we replace dAB with r and raise the above formula in the second power to get rid of the square root sign. Moreover, we can replace point A with the centre C of the circle. In this way, we obtain for the distance of a point B on the circle from the origin C the following equation:

(xB - xC )2 + (yB - yC )2 = r2

We denote the centre by C when it is not at the origin. When the centre of the circle corresponds to the origin, we denote this point by O.

Generalizing for any point (x, y) of the circle, we can write

(x - a)2 + (y - b)2 = r2

where (a, b) are the coordinates of the centre C of the circle.

The last equation is known as the equation of circle. As a special case of the equation of a circle, we can mention situations where the centre of the circle corresponds to the origin O(0, 0). In these cases, the equation of circle becomes

(x - 0)2 + (y - 0)2 = r2

or

x2 + y2 = r2

Example 1

Find the coordinates of the centre and the radius for the circles given by the following equations:

  1. x2 + y2 = 25
  2. x2 + (y + 3)2 = 9
  3. (x - 1)2 + y2 = 16
  4. (x - 2)2 + (y + 3)2 = 36

Solution 1

By default, we will assume the equation of any of the circles below having the general form

(x - a)2 + (y - b)2 = r2
  1. This is the simplest form of the equation of a circle as a = 0 and b = 0, so the centre is at O(0, 0).
    As for the radius, we have
    x2 + y2 = r2
    x2 + y2 = 25
    Thus, since r2 = 25, we obtain the radius:
    r = √25
    r = 5 units
    The figure below shows the above circle. Math Tutorials: Circle Graphs Example
  2. In this case, we have a = 0 and b = -3. Therefore, the centre is at C(0, -3) and the radius is
    r2 = 9
    r = √9
    r = 3 units
    The figure below shows the above circle. Math Tutorials: Circle Graphs Example
  3. In this case, we have a = 1 and b = 0. Therefore, the centre is at C(1, 0) and the radius is
    r2 = 16
    r = √16
    r = 4 units
    The figure below shows the above circle. Math Tutorials: Circle Graphs Example
  4. In this case, we have a = 2 and b = -3. Therefore, the centre is at C(2, -3) and the radius is
    r2 = 16
    r = √16
    r = 4 units

You have reached the end of Math lesson 15.6.2 Equation of a Circle. There are 5 lessons in this physics tutorial covering Circle Graphs, you can access all the lessons from this tutorial below.

More Circle Graphs Lessons and Learning Resources

Types of Graphs Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial		Revision
Notes		Revision
Questions
15.6Circle Graphs
Lesson IDMath Lesson TitleLessonVideo
Lesson
15.6.1Geometry Background
15.6.2Equation of a Circle
15.6.3Equation of a Circle when it is expressed as a Polynomial
15.6.4Finding the Equation of a Circle when a Graph is Provided
15.6.5What Else We Can Find Using the Equation of a Circle?

Whats next?

Enjoy the "Equation of a Circle" math lesson? People who liked the "Circle Graphs lesson found the following resources useful:

  1. Equation Feedback. Helps other - Leave a rating for this equation (see below)
  2. Types of Graphs Math tutorial: Circle Graphs. Read the Circle Graphs math tutorial and build your math knowledge of Types of Graphs
  3. Types of Graphs Revision Notes: Circle Graphs. Print the notes so you can revise the key points covered in the math tutorial for Circle Graphs
  4. Types of Graphs Practice Questions: Circle Graphs. Test and improve your knowledge of Circle Graphs with example questins and answers
  5. Check your calculations for Types of Graphs questions with our excellent Types of Graphs calculators which contain full equations and calculations clearly displayed line by line. See the Types of Graphs Calculators by iCalculator™ below.
  6. Continuing learning types of graphs - read our next math tutorial: Transforming Graphs

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