Calculate circles, arcs and sectors
Use circle circumference and area, then scale arc length and sector area by the angle's fraction of a full turn.
Section 6 · Mensuration
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which formula gives circumference from radius r?
What you will learn
- Calculate circumference and area of a circle
- Find arc length and sector area
- Give exact answers in terms of π when appropriate
Use radius consistently
Circumference C = 2πr = πd; area A = πr². A diameter is twice the radius, and circumference is a length while area is square units.
Cambridge supplies circle, arc and sector formulas in this section. Answers may be requested in terms of π; do not replace an exact answer with a decimal unless asked.
A sector is a fraction of the full circle
For central angle θ degrees, arc length = θ/360 × 2πr and sector area = θ/360 × πr². The Core syllabus restricts the sector angle to factors of 360°; Extended includes minor and major sectors.
A major sector uses the reflex angle greater than 180°. Check that the resulting arc is a fraction of the full circumference and the area a fraction of the circle.
Pause and explain
A 60° sector is what fraction of a full circle?
Worked example
For a circle of radius 7 cm, find its circumference, then the arc length and area of a 90° sector.
Show the worked solution
- Circumference = 2π(7) = 14π cm.
- A 90° angle is 90/360 = 1/4 of a full turn, so the arc is 14π/4 = 7π/2 cm.
- Circle area = 49π cm², so sector area = 49π/4 cm².
Answer Circumference 14π cm; arc 7π/2 cm; sector area 49π/4 cm².
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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.
A circle has radius 6 cm. Find its area and the arc length for a 120° sector.
Give me a hint
Use πr² for area and 120/360 of the circumference for the arc.
Compare my reasoning
- Area = π(6²) = 36π cm².
- Circumference = 12π cm.
- Arc length = 120/360 × 12π = 4π cm.
Area 36π cm²; arc length 4π cm.
Look for these in your work
- Radius squared for area
- Full circumference found
- Angle fraction applied
A circle has radius 5 cm. Find the area of a 216° sector in terms of π.
Give me a hint
Use the sector angle fraction 216/360.
Compare my reasoning
- 216/360 = 3/5.
- Full area = 25π cm².
- Sector area = 3/5 × 25π = 15π cm².
15π cm².
Look for these in your work
- Fraction simplified
- Circle area correct
- Sector fraction applied
A circular running track has radius 35 m. A runner completes a 72° arc. Find the exact distance and explain why the answer is a fraction of one lap.
Give me a hint
Compare the central angle with 360° and scale circumference.
Compare my reasoning
- The fraction of a lap is 72/360 = 1/5.
- Circumference = 2π(35) = 70π m.
- Arc distance = 1/5 × 70π = 14π m.
14π m; the runner travels one fifth of a full turn.
Look for these in your work
- Angle fraction identified
- Circumference found
- Length units retained
Common mistakes
- Using diameter in place of radius
- Squaring the circumference formula
- Scaling arc length by the square of the angle fraction
- Using a minor angle for a major sector
What can you explain now?
What is the area of a circle with radius 3 cm?
Compare with the explanation
9π cm²
Area = πr².
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How does the central angle determine arc length and sector area as fractions of a full circle?
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.