Use a degree fraction for arc length and sector area
Use the central angle as a fraction of 360 degrees · Distinguish the curved arc from sector perimeter · Keep area and boundary formulae separate
Section Paper 3 · Historical Core · calculator
Exam scope and assessment notes
Core and Extended foundations. 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
- Circle circumference and area
- Fractions of a whole turn
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
- Use the central angle as a fraction of 360 degrees
- Distinguish the curved arc from sector perimeter
- Keep area and boundary formulae separate
A fraction of a circle
For central angle θ degrees, arc length is (θ/360)2πr and sector area is (θ/360)πr². These degree formulae use a fraction of a full turn, not a radian formula. The stated central angle selects the sector; use 360−θ for its complementary sector if requested.
Include the straight edges when asked
For a non-degenerate sector with 0<θ<360, its perimeter is arc length plus two radii. The illustrated radius-6 cm sector has angle 120°: arc 4π cm, area 12π cm² and perimeter 12+4π cm. A complete circular region at 360° has circumference only; do not add two imaginary cuts.
Sources for this lesson
Pause and explain
What else belongs in the perimeter of a 90° sector, besides its arc?
Worked example
Find arc length, area and perimeter of a radius-6 cm, 120° sector.
Show the worked solution
- The angle fraction is 120/360 = 1/3.
- Arc=(1/3)12π=4π cm; area=(1/3)36π=12π cm²; add two radii for perimeter.
Answer Arc 4π cm; area 12π cm²; perimeter 12+4π cm.
Try a topic test
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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.
Find the arc length and area of a 90° sector of radius 8 cm.
Give me a hint
Use one quarter of the circle.
Compare my reasoning
- Arc=(1/4)16π = 4π cm.
- Area=(1/4)64π = 16π cm².
4π cm; 16π cm².
Look for these in your work
- Use the degree fraction
- Keep the quantities separate
Find the perimeter of a 60° sector of radius 6 cm.
Give me a hint
Add the two radii to one sixth of the circumference.
Compare my reasoning
- Arc=(1/6)12π = 2π cm.
- Perimeter=2π+6+6 = 12+2π cm.
12+2π cm.
Look for these in your work
- Count every boundary piece
- Retain an exact result
A full circular plate of radius 5 cm is compared with a semicircular plate of that radius. Find each perimeter.
Give me a hint
The full plate has no cut edges; the semicircle has a diameter edge.
Compare my reasoning
- Full circle perimeter=10π cm.
- Semicircle perimeter=5π+10 cm.
Full circle 10π cm; semicircle 10+5π cm.
Look for these in your work
- Use the actual region
- Do not add imaginary edges
Common mistakes
- An arc length alone is not sector perimeter.
- The angle must match the region being measured.
What can you explain now?
What else belongs in the perimeter of a 90° sector, besides its arc?
Compare with the explanation
Two radii.
They form its two straight boundary edges.
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Make it stick
What else belongs in the perimeter of a 90° sector, besides its arc?
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.