The Lesson

We can find the centre and the radius of a circle from the equation of a circle. Imagine we wanted to find the centre and radius of the circle with the equation:

How to Find the Centre and Radius from the Equation of a Circle

Finding the centre and radius from the equation of a circle is easy.

How to Find the Centre from the Equation of a Circle

Find the Cartesian coordinates of the centre of the circle from the equation.

Question

What is the centre of the circle with the equation (x − 1)2 + (y + 2)2 = 9?

Step-by-Step:

1

Find the brackets with the x in it.

2

Look at the number in these brackets and the sign in front of it.

In our example, it is − 1.

3

Change the sign of the answer (−1) to find the x-coordinate of the centre of the circle.
−1 → 1
The x-coordinate of the centre of the circle is 1.

4

Find the brackets with the y in it.

5

Look at the number in these brackets and the sign in front of it.

In our example, it is + 2.

6

Change the sign of the answer (+2) to find the y-coordinate of the centre of the circle.
+ 2 → − 2
The y-coordinate of the centre of the circle is −2.

Answer:

The centre of the circle with the equation (x − 1)2 + (y + 2)2 = 9 is:

How to Find the Radius from the Equation of a Circle

Find the radius of the circle from the equation.

Question

What is the radius of the circle with the equation (x − 1)2 + (y + 2)2 = 9?

Step-by-Step:

1

Find the number not in brackets. It is usually written to the right of the equals sign (=).

In our example, it is 9.

2

Find the square root of the answer (9).
√9 = 3

Answer:

The radius of the circle with the equation (x − 1)2 + (y + 2)2 = 9 is 3. The circle with the equation (x − 1)2 + (y + 2)2 = 9 has centre (1, −2) and radius 3.

Lesson Slides

The slider below shows another real example of how to find the centre and radius from the equation of a circle.

Top Tip

The Center of a Circle

The equation of a circle is:

The center is (a, b).
  • The number being subtracted from the x in the brackets is the x-coordinate of the center.
  • The number being subtracted from the y in the brackets is the y-coordinate of the center.
What if a coordinate of the center is negative? Imagine the center of the circle is (−1, 2). The equation will start:
(x − −1)2 + ...
Remember, that subtracting a negative number is the same as adding the positive number:
(x − −1)2 = (x + 1)2
A negative coordinate will have a + sign in front of it. A positive coordinate will have a − sign in front of it. With practise, it is straightforward to read off the center of the circle.

Beware

Equations That Don't Quite Look Right

Don't be confused if you see an equation which looks like this:
(x − 1)2 + (y − 3)2 − 49 = 0
This is still an equation of a circle, as can be seen with a little rearranging:
(x − 1)2 + (y − 3)2 = 49