Arc lengths and sector boundaries
Take the correct fraction of a circle and include straight edges when finding a sector perimeter.
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.
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.
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 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.
After trying it yourself, choose your next review. This is your self-assessment.
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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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources