Arc lengths and sector boundaries
Take the correct fraction of a circle and include straight edges when finding a sector perimeter.
Section Paper 3 · Core · Calculator
Core and Extended. Core sector angles are factors of 360°. Extended uses general angles and includes minor and major sectors; the major-sector transfer task below is Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Use circle circumference and area.
- Simplify fractions of 360°.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What fraction of a full turn is 90°?
What you will learn
- Calculate arc length and sector area.
- Distinguish minor and major sectors and their boundaries.
The central angle determines the fraction
A sector of central angle θ occupies θ/360 of a complete circle. Its arc length is (θ/360) × 2πr and its area is (θ/360) × πr². For radius 6 cm and angle 120°, the fraction is one third, giving arc 4π cm and area 12π cm². Arc length and sector area share a fraction but use different full-circle measures.
A minor sector uses the smaller central angle; the complementary major sector uses 360° minus that angle. Extended questions may use angles that do not divide 360 neatly. Retain the exact fraction and pi until the last calculation. A chord-bounded segment has a different area from a sector and needs subtraction of the enclosed triangle when applicable.
Trace every boundary edge
A sector's full perimeter is its curved arc plus two radii: L + 2r. For the 120° sector above it is 4π + 12 cm. A semicircle has curved length πr and perimeter πr + 2r. Do not take a fraction of the full circumference and call that the complete perimeter of the cut-out piece.
For a major sector, the same two radii form the straight edges, while the longer arc follows the major angle. In a joined compound shape, some of these straight edges may become internal and should be omitted from the outside perimeter. Mark the exposed boundary before adding lengths.
Pause and explain
A 90° sector has radius 8 cm. What is its full perimeter?
Worked example
Find the exact arc length, area and perimeter of a 120° sector with radius 6 cm.
Show the worked solution
- The fraction is 120/360 = 1/3, so arc length is (1/3) × 12π = 4π cm.
- Area is (1/3) × 36π = 12π cm².
- Add two radii for perimeter: 4π + 12 cm.
Answer Arc 4π cm; area 12π cm²; perimeter (4π + 12) cm.
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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 arc length and area of a 90° sector of radius 8 cm.
Give me a hint
Use one quarter of each full-circle measure.
Compare my reasoning
- Circumference is 16π cm, so the arc is 4π cm.
- Circle area is 64π cm², so sector area is 16π cm².
- Differentiate the length and area units.
4π cm; 16π cm².
Look for these in your work
- Used the same quarter fraction.
- Applied it to the correct full-circle measures.
A semicircle has radius 5 cm. Find its area and full perimeter.
Give me a hint
The diameter remains on the cut edge.
Compare my reasoning
- Area is half of 25π, or 25π/2 cm².
- The curved arc is 5π cm.
- Add the 10 cm diameter: perimeter is 5π + 10 cm.
Area 25π/2 cm²; perimeter (5π + 10) cm.
Look for these in your work
- Halved the circular area.
- Included the straight diameter edge.
Extended: a circle of radius 10 cm has a minor sector angle 72°. Find the remaining major sector's area and perimeter.
Give me a hint
The remaining angle is 360° − 72°.
Compare my reasoning
- The major angle is 288°, fraction 4/5.
- Its area is (4/5) × 100π = 80π cm².
- Its arc is (4/5) × 20π = 16π cm; add two radii to give 16π + 20 cm.
Area 80π cm²; perimeter (16π + 20) cm (Extended).
Look for these in your work
- Used the major angle.
- Included exposed radius edges.
Common mistakes
- Confusing an arc with a full perimeter.
- Using the minor angle for a major sector.
- Applying circumference units to sector area.
What can you explain now?
What must be added to a sector arc to obtain its full perimeter?
Compare with the explanation
Two radii.
The sector boundary contains the curved arc and both straight lines from its endpoints to the centre.
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Make it stick
Tomorrow, trace and label the boundary of a sector before calculating. Explain which edges count.
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.