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.

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Why Convert from General Form to Standard Form?

• In general form, A, B and C do not tell us anything about the circle.

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.

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

Question

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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.

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2

Consider the x 2 and x terms only.

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3

Replace the x 2 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.

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Write an x in the brackets.

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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.

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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.

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Divide the coefficient of x by 2.

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Square the halved coefficient (1).
1 2 = 1 × 1 = 1

Write it in the gap after the − sign.

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4

Consider the y 2 and y terms only.

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5

Replace the y 2 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.

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Write a y in the brackets.

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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.

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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.

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Divide the coefficient of y by 2.

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Square the halved coefficient (2).
2 2 = 2 × 2 = 4

Write it in the gap after the − sign.

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6

Add the numbers outside of the brackets together.

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7

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.

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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.