Write and read the equation of a circle with centre at the origin or at (a, b), and test whether a point is on, inside or outside it. It is Higher only, aimed at grade 8. Free questions with answers are on this page.
Objectives
Read the centre and radius from an equation.
Write the equation from a centre and a radius or a point.
Decide whether a point is on, inside or outside the circle.
Key points
Recognise x2+y2=r2 as a circle with centre (0,0) and radius r.
Read the centre and radius from (x−a)2+(y−b)2=r2, setting each bracket to zero to find the centre.
Write the equation from a centre and a radius, a centre and a point on the circle, or the ends of a diameter.
Substitute a point and compare with r2 to decide whether it is on, inside or outside the circle.
Worked examples
Example 1
Write the equation of the circle with centre (0, 0) and radius 7.
2 marks
The centre is the origin, so the equation is x^2 + y^2 = r^2
The radius squared is 7 x 7 = 49
x^2 + y^2 = 49
Answer: x^2+y^2=49
Common slips
Slip: (x - 7)^2 + (y - 7)^2 = r^2 The radius is not a coordinate of the centre.
Slip: The radius squared is 14 Square 7. Do not double it.
Slip: x^2 + y^2 = 7 The right-hand side is r squared, not r.
Example 2
A circle has centre (3, -2) and radius 5. Write its equation.
2 marks
Subtract the centre: (x - 3) and (y + 2)
The radius squared is 5 x 5 = 25
(x - 3)^2 + (y + 2)^2 = 25
Answer: (x-3)^2+(y+2)^2=25
Common slips
Slip: (x + 3) and (y - 2) Subtracting -2 is the same as adding 2.
Slip: The radius squared is 5 Square the radius before you write it.
Slip: (x - 3)^2 + (y + 2)^2 = 5 The equation finishes with r squared.
Example 3
The points A(2, 1) and B(8, 9) are the ends of a diameter. Write the equation of the circle.
3 marks
The centre is the midpoint of AB, which is (5, 5)
The steps from the centre to A are 3 and 4, so r squared is 9 + 16 = 25
(x - 5)^2 + (y - 5)^2 = 25
Answer: (x-5)^2+(y-5)^2=25
Common slips
Slip: The centre is A, which is (2, 1) A is on the circle. The centre is half way along the diameter.
Slip: r squared is 3 + 4 = 7 Square each step before adding.
Slip: (x - 5)^2 + (y - 5)^2 = 5 The right-hand side is r squared, which is 25.
Questions and answers
GCSE Maths equation of a circle questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Write the equation of the circle with centre (0, 0) and radius 6.
Show answer
Answer: x^2+y^2=36
The centre is the origin.
Square 6 to get 36.
M1 — The centre is the origin.
A1 — Answer x^2+y^2=36
Question 2 Easy2 marks
What is the radius of the circle x^2 + y^2 = 81?
Show answer
Answer: 9
The right-hand side is r squared.
Square-root 81.
M1 — The right-hand side is r squared.
A1 — Answer 9
Question 3 Medium2 marks
Write the equation of the circle with centre (1, 2) and radius 3.
Show answer
Answer: (x-1)^2+(y-2)^2=9
Subtract the centre in each bracket.
The radius squared is 9.
M1 — Subtract the centre in each bracket.
A1 — Answer (x-1)^2+(y-2)^2=9
Question 4 Medium2 marks
A circle has centre (-4, 3) and radius 2. Write its equation.
Show answer
Answer: (x+4)^2+(y-3)^2=4
Subtracting -4 gives x + 4.
The radius squared is 4.
M1 — Subtracting -4 gives x + 4.
A1 — Answer (x+4)^2+(y-3)^2=4
Question 5 Hard2 marks
Is the point (6, 4) inside, on or outside the circle (x - 2)^2 + (y - 1)^2 = 25? Answer inside, on or outside.
Show answer
Answer: on
Substitute: 16 + 9.
Compare 25 with 25.
M1 — Substitute: 16 + 9.
A1 — Answer on
Question 6 Hard2 marks
Is the origin inside, on or outside the circle (x - 3)^2 + (y - 4)^2 = 16? Answer inside, on or outside.
Show answer
Answer: outside
Substitute (0, 0): 9 + 16 = 25.
Compare 25 with 16.
M1 — Substitute (0, 0): 9 + 16 = 25.
A1 — Answer outside
Question 7 Exam-style4 marks
A circle has centre (1, -2) and passes through the point (4, 2). Write the equation of the circle.
Show answer
Answer: (x-1)^2+(y+2)^2=25
The steps from the centre to the point are 3 and 4.
r squared is 9 + 16 = 25.
Subtracting -2 gives y + 2.
The equation is (x - 1)^2 + (y + 2)^2 = 25.
M1 — Finds the steps 3 and 4 from the centre to the point
M1 — Squares and adds them to get r squared
A1 — r squared is 25
A1 — Writes (x - 1)^2 + (y + 2)^2 = 25
Common mistakes
Most lost marks on this topic are one slip.
Reading the centre straight from the signs in the brackets
Set each bracket to zero and solve to find the centre
Writing the radius instead of the radius squared on the right
Square the radius first, or read the number on the right as r squared
Adding the steps from the centre without squaring them
Square the horizontal and vertical steps, then add
Spot the slip
A student writes the circle with centre (3, 2) and radius 4 as (x + 3)^2 + (y + 2)^2 = 4. What went wrong?
The signs are reversed and the 4 should be 16
The centre should be (-3, -2)
Nothing, the equation is right
Show the slip
The signs are reversed and the 4 should be 16 The brackets subtract the centre, and the right-hand side is r squared: (x - 3)^2 + (y - 2)^2 = 16.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.