Classify how two circles meet
Compare the distance between centres with the sum and difference of the radii to classify circle intersections.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which values do you compare with the centre distance?
What you will learn
- Find the distance between two circle centres
- Classify two circles as intersecting, touching or disjoint
- Distinguish external and internal tangency and containment
The centre distance controls the intersection type
Let the radii be r₁ and r₂, and let d be the distance between centres. If |r₁ − r₂| < d < r₁ + r₂, the circles cross at two points.
If d = r₁ + r₂, they touch externally. If d = |r₁ − r₂| > 0, they touch internally.
If d > r₁ + r₂, the circles are separate. If 0 ≤ d < |r₁ − r₂|, one is inside the other without touching. If d = 0 and radii match, the circles coincide.
Calculate before deciding
Use the distance formula for centres (a₁, b₁) and (a₂, b₂): d = √((a₂ − a₁)² + (b₂ − b₁)²). Compare d with both r₁ + r₂ and |r₁ − r₂|.
Equality gives a tangent contact; strict inequalities between the two thresholds give two intersections. State the geometric reason, not only the final label.
Pause and explain
Two radii are 7 and 3, and the centres are 4 units apart. What happens?
Worked example
A circle of radius 5 has centre (0, 0); another of radius 3 has centre (8, 0). How do they meet?
Show the worked solution
- The centre distance is d = 8.
- The sum of radii is 8 and their difference is 2.
- Since d = r₁ + r₂, the circles touch externally at one point.
Answer They touch externally because d = 8 = 5 + 3.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Classify circles with centres (0, 0) and (8, 0), both of radius 5.
Give me a hint
Compare d = 8 with the radius difference and sum.
Compare my reasoning
- The centre distance is 8.
- The radius difference is 0 and the sum is 10.
- 0 < 8 < 10, so they intersect at two points.
Two intersections.
Look for these in your work
- Centre distance found
- Both thresholds compared
- Intersection count justified
A radius-4 circle is centred at (1, 2). A radius-3 circle is centred at (10, 2). Do they meet?
Give me a hint
Find the centre distance and compare with 4 + 3.
Compare my reasoning
- The centre distance is 9.
- The radii sum to 7.
- Since 9 > 7, the circles are disjoint.
They do not intersect; the centre distance 9 exceeds the radius sum 7.
Look for these in your work
- Distance calculated
- External threshold found
- No-intersection result justified
A radius-7 circle and a radius-3 circle have centres 2 units apart. State their relationship and why there are no common points.
Give me a hint
Compare 2 with the difference of their radii.
Compare my reasoning
- The radius difference is 7 − 3 = 4.
- The centre distance 2 is less than 4.
- The smaller circle is wholly inside the larger and does not touch it.
No intersection; one circle lies inside the other because 2 < 4.
Look for these in your work
- Radius difference found
- Distance compared
- Containment explained
Common mistakes
- Comparing centre distance with only one radius
- Calling d = r₁ + r₂ an intersection at two points
- Treating an inner contained circle as intersecting
- Forgetting coincident circles when centres and radii both match
What can you explain now?
If |r₁ − r₂| < d < r₁ + r₂, how many intersection points are there?
Compare with the explanation
Two
The centre distance lies strictly between the internal and external tangency thresholds.
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Make it stick
What centre-distance comparisons distinguish two intersections, tangency and no intersection?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.