Converting an Equation of a Circle from General to Standard Form
(KS3, Year 8)

We can convert an equation of a circle from general form to standard form.

circle equation convert general to standard

Why Convert from General Form to Standard Form?

The same circle can be written in general form and standard form. The two forms of equations of a circle tell us different things about the circle.
  • In general form, A, B and C do not tell us anything about the circle.

    circle equation general no meaning
  • In standard form, • a and b tells us that (a, b) are the Cartesian coordinates of the center of the circle. • r tells us the radius of the circle. circle_equation_standard_meaning
If we are given a circle in general form, we can convert it to standard form to understand more about the circle.

A Real Example of How to Convert an Equation of a Circle from General to Standard Form

Question

Convert the equation of a circle in general form shown below into standard form. Find the center and radius of the circle.

Step-by-Step:

1

Group the x's and y's together.

circle equations convert general to standard step 1

2

Consider the x2 and x terms only.

circle equations convert general to standard step 2

3

Complete the square on these terms.
  • Replace the x2 and x terms with a squared bracket. Leave a gap inside the brackets for two terms. Leave a gap after the brackets for a number to be subtracted.

    circle equations convert general to standard step 3 1
  • Write an x in the brackets.

    circle equations convert general to standard step 3 2
  • Look at the original equation. Find the sign in front of the x term. In our example, it is . Write this sign after the x in the brackets.

    circle equations convert general to standard step 3 3
  • Look at the original equation. Find the number in front of the x term (called the coefficient of x). In our example, it is 2.

    circle equations convert general to standard step 3 4 Divide the coefficient of x by 2.
    2 ÷ 2 = 1
  • Write the halved coefficient (1) in the gap in the brackets.

    circle equations convert general to standard step 3 5
  • Square the halved coefficient (1).
    12 = 1 × 1 = 1
    Write it in the gap after the sign.

    circle equations convert general to standard step 3 6
We have completed the square on these terms.

circle equations convert general to standard complete the square x

4

Consider the y2 and y terms only.

circle equations convert general to standard step 4

5

Complete the square on these terms.
  • Replace the y2 and y terms with a squared bracket. Leave a gap inside the brackets for two terms. Leave a gap after the brackets for a number to be subtracted.

    circle equations convert general to standard step 5 1
  • Write a y in the brackets.

    circle equations convert general to standard step 5 2
  • Look at the original equation. Find the sign in front of the y term. In our example, it is . Write this sign after the y in the brackets.

    circle equations convert general to standard step 5 3
  • Look at the original equation. Find the number in front of the y term (called the coefficient of y). In our example, it is 4.

    circle equations convert general to standard step 5 4 Divide the coefficient of y by 2.
    4 ÷ 2 = 2
  • Write the halved coefficient (2) in the gap in the brackets.

    circle equations convert general to standard step 5 5
  • Square the halved coefficient (2).
    22 = 2 × 2 = 4
    Write it in the gap after the sign.

    circle equations convert general to standard step 5 6
We have completed the square on these terms.

circle equations convert general to standard complete the square y

6

Add the numbers outside of the brackets together.

circle equations convert general to standard step 6 1
− 1 + − 4 + − 4 = − 1 − 4 − 4 = − 9
circle equations convert general to standard step 6 2

7

Rearrange the equation so the number is on the right hand side of the equals sign (=).
(x − 1)2 + (y − 2)2 − 9 = 0
(x − 1)2 + (y − 2)2 − 9 + 9= 0 + 9 Add 9 to boths sides
(x − 1)2 + (y − 2)2 = 9

Answer:

We have converted x2 + y2 − 2x − 4y − 4 = 0 (in general form) to (x − 1)2 + (y − 2)2 = 9 (in standard form). This is a circle centred at (1, 2) with a radius of 9.


how to find the centre and radius from the equation of a centre

Lesson Slides

The slider below gives a real example of how to convert an equation of a circle from general form to standard form.
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This page was written by Stephen Clarke.