Math Lesson 15.6.4 - Finding the Equation of a Circle when a Graph is Provided

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Welcome to our Math lesson on Finding the Equation of a Circle when a Graph is Provided, this is the fourth 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.

Finding the Equation of a Circle when a Graph is Provided

As always when dealing with graphs, the study would be incomplete if the reverse scenario is not discussed. So far, we have plotted a circle in a coordinates system when the equation was given (or almost given). Now, we are going to see the reverse situation, i.e. when a circle is shown in the coordinates system and we want to find the equation of this circle. Let's use the figure below as a reference.

The procedure to find the equation of a circle when the graph is shown is as follows.

Step 1 - The first thing to do is to detect the four extreme points of the circle: the uppermost, the lowermost, the leftmost and the rightmost ones. We can denote these points by A, B, C and D if we wish to. In the above figure, we have A(-1, -1), B(4, -4), C(9, 1) and D(4, 6). In this way, we obtain two crossing segments that represent two perpendicular diameters: AC and BD.

Step 2 - Using the midpoint formula for each segment to identify the centre C. Obviously, we need to use twice this formula, one for the horizontal and the other for the vertical direction. In our case, we have

and

This means the centre of the circle is C(4, 1).

Step 3 - Calculate the radius by dividing the change in coordinates of any of the above diameters by 2 (no matter which direction we choose for this, as all radii of a circle are equal in length). For example, considering the y-coordinates we obtain

Step 4 - Use the values found in the previous steps to write the equation of the unknown circle in the form (x - a)² + (y - b)² = r². In our case, we have for the equation of the given circle

because r² = 5² = 25, and a = 4 and b = 1 are the coordinates of the centre C.

Let's consider an example to clarify this point.

Example 3

Find the equation of the circle shown in the figure below (in both forms).

Solution 3

It is easy to see that in the horizontal direction the circle lies between x_min = -2 and x_max = 10 while in the vertical direction it lies between y_min = -5 and y_max = 7. Therefore, we can identify the centre C of the circle by applying the equation of the segment midpoint for each direction separately.

and

Therefore, the centre C(a, b) is at C(4, 1).

As for the radius, we can find it by using any of the directions, for example, the horizontal one. Given that the difference between the maximum and minimum coordinate gives the diameter, and since the radius is half of the diameter, we have for the radius r:

Hence, given that the general form of the equation of the circle shown in the graph is

Hence, substituting the known values yields

or

Expanding the last equation yields the other form of the circle's equation.

Remark! The study of circles in the coordinates system does not end here. We will get back top this topic in the Conics chapter, which much more things to learn about circles. Hence, consider this tutorial as a kind of introduction to circles.

You have reached the end of Math lesson 15.6.4 Finding the Equation of a Circle when a Graph is Provided. There are 5 lessons in this physics tutorial covering Circle Graphs, you can access all the lessons from this tutorial below.

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