Reverse sector data and measure annular sector boundaries
Find a degree angle from arc data · Subtract sectors with the same central angle · Account for both curved edges and radial strips
Section Paper 2 · Historical Extended · non-calculator
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Simple arcs and sectors
- Algebraic rearrangement
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Find a degree angle from arc data
- Subtract sectors with the same central angle
- Account for both curved edges and radial strips
Reverse the degree fraction
From an arc L and radius r, θ=360L/(2πr). The ratio is a fraction of the circumference. Check the indicated angle range: an ordinary single sector cannot have an angle greater than 360°. The complementary arc uses 360−θ, not the same fraction again.
An annular sector
With outer radius R, inner radius r and common angle θ, area is (θ/360)π(R²−r²). For 0<θ<360, perimeter includes the outer arc, inner arc and two radial lengths R−r. The area difference is not π(R−r)². A complete annulus has only its two circular boundaries.
Sources for this lesson
Pause and explain
Which expression gives the area of a complete annulus?
Worked example
An arc is 5π cm long on a circle of radius 6 cm. Find its degree angle.
Show the worked solution
- The full circumference is 12π cm.
- θ=360×5π/(12π) = 150°.
Answer 150°.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
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.
An arc on a radius-9 cm circle has length 6π cm. Find its degree angle.
Give me a hint
Compare with circumference 18π.
Compare my reasoning
- Arc fraction=6π/(18π)=1/3.
- θ=(1/3)360 = 120°.
120°.
Look for these in your work
- Use the correct circumference
- Check the angle range
A 90° annular sector has outer radius 8 cm and inner radius 4 cm. Find its area and perimeter.
Give me a hint
Use a quarter of both circles and two radial strips.
Compare my reasoning
- Area=(1/4)π(64−16)=12π cm².
- Arcs total 4π+2π=6π cm; radial edges total 8 cm.
Area 12π cm²; perimeter 8+6π cm.
Look for these in your work
- Subtract the correct regions
- Include both curved boundaries
A radius-5 cm circle has a 72° sector marked. Find the complementary sector area and perimeter.
Give me a hint
Its angle is 288°, or four fifths of a turn.
Compare my reasoning
- Area=(4/5)25π=20π cm².
- Arc=(4/5)10π=8π cm; add two radii.
Area 20π cm²; perimeter 10+8π cm.
Look for these in your work
- Select the requested sector
- Use its actual boundary
Common mistakes
- An inner arc is still an exposed boundary of an annular sector.
- Subtract squared radii for area, not square their difference.
What can you explain now?
Which expression gives the area of a complete annulus?
Compare with the explanation
π(R²−r²).
Subtract the inner disk area from the outer disk area.
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Make it stick
Which expression gives the area of a complete annulus?
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.