6.2PureCore

Equation of a Circle

A circle is the set of points a fixed distance (the radius) from a centre. That single idea, combined with Pythagoras, gives the equation of any circle — and lets you read off its centre and radius at a glance.

30 min Circles

Edexcel AS Level Maths: 6.2 Equation of a CircleCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Write the equation of a circle from its centre and radius
Read the centre and radius from a circle equation
Find the centre and radius by completing the square
Determine whether a point lies on, inside or outside a circle

The standard form

A circle with centre $(a, b)$ and radius $r$ has a beautifully simple equation, derived straight from Pythagoras.

(x - a)^{2} + (y - b)^{2} = r^{2}

Centre

(a, b)

, radius

r

. Watch the signs: the centre coordinates are subtracted.

Example 1Write the equation of a circle with centre

(2, -3)

and radius

5

1Substitute

a = 2

b = -3

r = 5

2Note

-(-3) = +3

inside the bracket.

Answer

(x - 2)^{2} + (y + 3)^{2} = 25

Tip — The centre coordinates appear with the OPPOSITE sign inside the brackets: (x − a) means centre x = a.

Reading off centre and radius

Given the standard form, the centre is read directly (with sign flips) and the radius is the square root of the right-hand side.

Example 1State the centre and radius of

(x + 1)^2 + (y - 4)^2 = 9

1Centre: flip signs →

(-1, 4)

2Radius:

\sqrt{9} = 3

Answercentre

(-1, 4)

, radius

3

Completing the square form

Circles are often given expanded, like $x^2 + y^2 + 2x - 6y - 6 = 0$ . Complete the square on the $x$ terms and on the $y$ terms to recover the standard form.

Example 1Find the centre and radius of

x^2 + y^2 + 2x - 6y - 6 = 0

1Group:

(x^2 + 2x) + (y^2 - 6y) = 6

2Complete the square:

(x + 1)^2 - 1 + (y - 3)^2 - 9 = 6

3Tidy:

(x + 1)^2 + (y - 3)^2 = 16

Answercentre

(-1, 3)

, radius

4

Tip — Move the constant to the right and remember to add back the numbers created by completing each square.

Formula recap

(x - a)^{2} + (y - b)^{2} = r^{2}

Centre (a, b), radius r.

\text{centre} = (a, b), \;\; r = \sqrt{\text{RHS}}

Reading the circle.

Common mistakes to avoid

Reading the centre of (x + 1)² + (y − 4)² = 9 as (1, −4).

Flip the signs: the centre is (−1, 4).

Taking the radius as the right-hand side instead of its square root.

r² = 9 means r = 3, not 9.

Key takeaways

Standard form: (x − a)² + (y − b)² = r².
Centre coordinates have opposite signs to the brackets; radius = √(RHS).
For an expanded equation, complete the square in x and y to find centre and radius.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Equation of a Circle. Use the feedback to decide what needs another look.

15 past-paper questions · 34 marks

Practise this topic

Related lessons

All 5 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.