GCSE & IGCSE maths practice

Equation of a circle questions and answers

A circle with centre the origin and radius $r$ has the equation $x^2 + y^2 = r^2$ . So $x^2 + y^2 = 25$ has radius 5. A tangent touches the circle at one point and is perpendicular to the radius there: find the radius's gradient, flip it and change the sign, then use the point.

8 questions · Higher tier · Every answer is checked as you type it, with a full worked solution. Updated 27 Oct 2026.

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Where equation of a circle comes up

GCSE (AQA, Edexcel, OCR)	Higher only: recognise and use the equation of a circle with centre at the origin, and find the equation of a tangent to a circle at a given point.
Cambridge IGCSE (0580)	Not in the 0580 syllabus (the equation of a circle isn't listed).

The method

x² + y² = r²: the radius is the square root of the number.
A point lies on the circle if its coordinates fit the equation.
Tangent at (a, b): radius gradient b ÷ a; tangent gradient −a ÷ b.
Then y − b = m(x − a).

Equation of a circle questions

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Warm-up

Question 1Warm-upHigher

What is the radius of the circle $x^2 + y^2 = 25$ ?

Question 2Warm-upHigher

A circle has centre $(0, 0)$ and radius 7. Complete its equation.

x² + y² =

Exam-style

Question 3Exam-styleHigher

Find the radius of $x^2 + y^2 = 20$ in surd form.

Question 4Exam-styleHigher

Does the point $(3, -4)$ lie on the circle $x^2 + y^2 = 25$ ?

Question 5Exam-styleHigher

The point $(-6, k)$ lies on $x^2 + y^2 = 100$ , with $k > 0$ . Find $k$ .

k =

Question 6Exam-styleHigher

$P(3, 4)$ is on the circle $x^2 + y^2 = 25$ . What is the gradient of the radius $OP$ ?

Challenge

Question 7ChallengeHigher

What is the gradient of the tangent to $x^2 + y^2 = 25$ at $(3, 4)$ ?

Question 8ChallengeHigher

Find the equation of the tangent to $x^2 + y^2 = 25$ at the point $(3, 4)$ .

y =

Common mistakes

Giving r² as the radius (x² + y² = 25 has radius 5, not 25).
Using the radius's gradient for the tangent.
Losing the sign of a negative coordinate.

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Sources

The questions are our own. The “where it comes up” notes are from each board's published specification, checked on 26 October 2026:

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