## 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?

# 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.

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?

# 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

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.

## 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.

## 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