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?

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

circle equations convert general to standard example

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.

2

Consider the x 2 and x terms only.

3

x 2

Leave a gap

after the brackets

circle equations convert general to standard step 3 1

circle equations convert general to standard step 3 2

−

circle equations convert general to standard step 3 3

circle equations convert general to standard step 3 4

Divide the coefficient of x by 2.

circle equations convert general to standard step 3 5

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 y 2 and y terms only.

circle equations convert general to standard step 4

5

y 2

Leave a gap

after the brackets

circle equations convert general to standard step 5 1

circle equations convert general to standard step 5 2

−

circle equations convert general to standard step 5 3

circle equations convert general to standard step 5 4

Divide the coefficient of y by 2.

circle equations convert general to standard <a href= step 5 5" width="521" height="119" />

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 x 2 + y 2 − 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.

circle equations convert general to standard answer

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.

This page was written by Stephen Clarke.

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