Equation from the circumference
A circle has centre (0, 0) and circumference 36π. Work out the equation of the circle.
- 2πr = 36π, so r = 18
- r² = 324
Answer: x² + y² = 324
Equation of a circle, tangents & intersections
Coordinate Circle Laboratory
Built from the circle questions on recent AQA and Edexcel GCSE Higher papers: write the equation from a circumference or a point, find the equation of a tangent, find where a line meets a circle, and translate a circle by a vector.
Circle equations are GCSE Higher only (AQA and Edexcel A16, A19). They are not on Cambridge IGCSE 0580, but the tangent and intersection stations practise IGCSE Extended skills: perpendicular gradients (E3.7) and solving a linear and a non-linear equation together (E2.5).
What students practise
- Radius and equation of x² + y² = r², including surd radii
- Equations from a radius, diameter, graph, circumference, area or a point
- Points on, inside or outside a circle; missing coordinates
- Where a circle meets the axes; the other end of a diameter; chord lengths
- Gradient of the radius and of the tangent
- Equation of a tangent at a point, and finding the point of contact from a tangent
- Where a line meets a circle (linear and quadratic simultaneous equations)
- Translating a circle by a vector; multi-step exam problems
How it works
Learn
10 lessons, each labelled with the exams it is for, with interactive models and worked examples.
Practise
Five stations, six fresh questions each visit, with hints that fade.
Challenge
Mixed questions from every station with a skill-by-skill report.
For teachers
Open Teacher options to choose Step by step (one skill at a time) or Exam level (surds, tangent equations, multi-step problems). Nothing is uploaded: progress stays in each student's browser, so no accounts or data-protection paperwork are needed.
Which exams it covers
Circle equations are GCSE Higher content on every board. They are not on the Cambridge IGCSE 0580 syllabus, though the algebra behind tangents and intersections is.
| Topic | GCSE (AQA, Edexcel) | Cambridge IGCSE 0580 |
|---|---|---|
| Equation of a circle centred at the origin, x² + y² = r² | A16 (Higher) | Not on 0580 |
| Equation of a tangent to a circle at a given point | A16 (Higher) | Not on 0580 (uses E3.7 perpendicular gradients) |
| Where a line meets a circle: linear and quadratic simultaneous equations | A19 (Higher) | E2.5 (the algebra, without circles) |
| Translating a graph by a vector | A13 (Higher) | Not on 0580 |
Worked examples
Four examples in the style of recent GCSE Higher papers.
Tangent at a point
Find the equation of the tangent to x² + y² = 25 at the point (3, 4). Give your answer in the form y = mx + c.
- Gradient of the radius to (3, 4) = 4/3
- The tangent is perpendicular: gradient −3/4
- y − 4 = −3/4(x − 3)
Answer: y = −3/4x + 25/4
Point of contact
The line 2x + y = 25 is the tangent to a circle with centre (0, 0) at the point P. Find P.
- The radius OP is perpendicular to the tangent, so P = (2k, k)
- 2(2k) + k = 25, so 5k = 25 and k = 5
Answer: P(10, 5)
Line and circle
Find where the line y = x + 1 meets the circle x² + y² = 25.
- Substitute: x² + (x + 1)² = 25
- 2x² + 2x − 24 = 0, so x² + x − 12 = 0
- (x + 4)(x − 3) = 0, so x = −4 or x = 3
Answer: (−4, −3) and (3, 4)
Common mistakes the game targets
- Using r² as the radius. In x² + y² = 49 the radius is √49 = 7, not 49.
- Using the radius gradient for the tangent. The tangent is perpendicular to the radius, so use the negative reciprocal.
- Squaring negatives without brackets. For (−3, 4), r² = (−3)² + 4² = 25. Typing −3² on a calculator gives −9.
- Finding y from the circle instead of the line. After solving for x, substitute into the line to get the matching y; the circle gives two y-values.
- Forgetting the radius stays the same. Translating a circle moves its centre by the vector; the radius does not change.
Questions teachers and parents ask
Is the game free?
Yes. It is free to use, with no account, no adverts and no tracking.
Is it on the IGCSE syllabus?
No. Circle equations are not in Cambridge IGCSE 0580; this game is for GCSE Higher. IGCSE Extended students can still use the tangent and intersection stations to practise perpendicular gradients and simultaneous equations.
Can students type exact answers?
Yes. Surds can be typed as 2sqrt5, points as (3, −4), lines in any form such as y = −3/4x + 25/4 or 3x + 4y = 25, and circles as x^2 + y^2 = 25.
Can I change the difficulty?
Yes. Teacher options switch between Step by step and Exam level.
Can students print their results?
Yes. The Challenge report has a Print report button.
Need worksheets and lessons to go with the games?
Printable IGCSE and GCSE resources from Mathwright: worksheets, exam-style practice and full worked solutions.
More coordinate geometry games
Coordinates & the four quadrants
Coordinate Treasure Map
Travel from Base Camp to the Treasure Chest by reading, plotting and moving between coordinates in all four quadrants.
Midpoint & length of a line segment
Coordinate Rescue Mission
Six stations on the midpoint and length of a line segment, missing points, exact surd lengths and dividing a line, with past-paper question types.
Gradient & straight-line graphs
Gradient City
Six stations covering every straight-line question on IGCSE and GCSE papers: gradient, drawing graphs, y = mx + c, parallel and perpendicular lines, and real-life gradients.