Equation of a circle: (x − a)² + (y − b)² = r²
The equation of a circle is (x − a)² + (y − b)² = r².
What each letter means
- \((a,\ b)\) the centre of the circle
- \(r\) the radius
- \((x,\ y)\) any point on the circle
When to use it
To write down a circle from its centre and radius, or to read the centre and radius from an equation (complete the square first if it is expanded).
Worked example
Find the centre and radius of the circle \(x^2+y^2-6x+4y-12=0\).
- \((x-3)^2-9+(y+2)^2-4-12=0\)
- \((x-3)^2+(y+2)^2=25\)
Answer: Centre \((3,-2)\), radius \(5\)
Common mistake
Getting the signs of the centre wrong: (x + 3)² means a = −3.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | Not in the Edexcel booklet: learn it |
| Edexcel International A Level | Not in the booklet: learn it |
| AQA or OCR A Level | Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page |
From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).
Practise and revise
- Practise A Level Maths practice
- Revise Coordinate geometry
- Revise IAL P2 circles
- Print Edexcel A Level (9MA0) one-page formula sheet
Questions
What is the equation of a circle formula?
The equation of a circle is (x − a)² + (y − b)² = r². (a, b): the centre of the circle; r: the radius; (x, y): any point on the circle.
Is the equation of a circle given in the exam?
Edexcel A Level (9MA0): not in the Edexcel booklet: learn it. Edexcel International A Level: not in the booklet: learn it. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.
How do I find the centre from the expanded form?
Complete the square in x and in y. For x² + y² + 2gx + 2fy + c = 0 the centre is (−g, −f) and the radius is √(g² + f² − c).
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.