Equation of a circle Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Equation of a circle questions. See exactly how to solve problems on equation of a circle, centre the origin, radius from equation, square roots.

equation of a circlecentre the originradius from equationsquare rootscentre of a circleorigin
GCSE Higher70 questionsStep-by-step solutions
Question 1
1 markeasy
Write down the equation of the circle with centre the origin and radius 77.

Worked solution

  1. Recall the standard form

    x2+y2=r2x^2 + y^2 = r^2

    A circle with centre the origin (0, 0) and radius r always has this equation.

  2. Square the radius

    r=7⇒r2=72=49r = 7 \Rightarrow r^2 = 7^2 = 49

    The equation uses r squared, not r, so square the 7 first.

  3. Write the equation

    x2+y2=49x^2 + y^2 = 49

    The circle of radius 7 centred on the origin is x2+y2=49x^2 + y^2 = 49.

Answer
x2+y2=49x^2 + y^2 = 49
Question 2
1 markeasy
A circle has equation x2+y2=36x^2 + y^2 = 36. Write down the radius of the circle.

Worked solution

  1. Compare with the standard form

    x2+y2=r2andx2+y2=36x^2 + y^2 = r^2 \quad\text{and}\quad x^2 + y^2 = 36

    Matching the two equations shows that r2=36r^2 = 36.

  2. Square root to undo the square

    r=36=6r = \sqrt{36} = 6

    A radius is a length, so take the positive square root only.

  3. State the radius

    r=6r = 6

    The circle has radius 6 units.

Answer
66
Question 3
1 markeasy
A circle has equation x2+y2=81x^2 + y^2 = 81. Write down the coordinates of the centre of the circle.

Worked solution

  1. Recognise the form

    x2+y2=81x^2 + y^2 = 81

    There is nothing added to or subtracted from x or y inside the squares.

  2. Read off the centre

    x2+y2=r2⇒centre (0, 0)x^2 + y^2 = r^2 \Rightarrow \text{centre } (0,\, 0)

    Every equation of the form x2+y2=r2x^2 + y^2 = r^2 is centred on the origin.

  3. State the centre

    (0, 0)(0,\, 0)

    The centre of this circle is the origin, (0, 0).

Answer
(0,0)(0, 0)
Question 4
1 markeasy
Which one of these equations represents a circle with centre the origin?

Worked solution

  1. Recall what a circle equation looks like

    x2+y2=r2x^2 + y^2 = r^2

    Both letters are squared, both squares are ADDED, and the right-hand side is a positive constant.

  2. Rule out the others

    x2−y2=30,y=x2+30,x+y=30,x2+y=30x^2 - y^2 = 30,\quad y = x^2 + 30,\quad x + y = 30,\quad x^2 + y = 30

    A subtraction of squares is not a circle; y=x2+30y = x^2 + 30 and x2+y=30x^2 + y = 30 are parabolas; x+y=30x + y = 30 is a straight line.

  3. Choose the circle

    x2+y2=30x^2 + y^2 = 30

    Only x2+y2=30x^2 + y^2 = 30 fits x2+y2=r2x^2 + y^2 = r^2 — it is a circle of radius sqrt(30) centred on the origin.

Answer
x2+y2=30x^2 + y^2 = 30
Question 5
2 markseasy
The point (3,4)(3, 4) lies on a circle with centre the origin. Work out the radius of the circle.

Worked solution

  1. Use the equation of the circle

    r2=x2+y2=32+42r^2 = x^2 + y^2 = 3^2 + 4^2

    The point (3, 4) lies on the circle, so its coordinates satisfy x2+y2=r2x^2 + y^2 = r^2.

  2. Work out r squared

    32+42=9+16=253^2 + 4^2 = 9 + 16 = 25

    This is just Pythagoras: the radius is the hypotenuse of a 3-4-5 triangle.

  3. Square root to find the radius

    r=25=5r = \sqrt{25} = 5

    The radius is 5 units.

Answer
55

Unlock 65 more Equation of a circle questions

Create a free account to work through every GCSE Equation of a circle question with instant step-by-step worked solutions, progress tracking and interactive lessons.

  • Full worked solutions for every question
  • Interactive lessons and instant feedback
  • Track your mastery across every topic
Create a Free Account

No card required · Free forever

More Equation of a circle practice

Hard QuestionsPractice Questions

Related Algebra topics

Algebraic fractionsCoordinatesCubic and reciprocal graphsExponential and trig graphsExpanding binomialsFactorising ax2 + bx + c