Find sector areas and perimeters
Use radian formulas for sector area and arc length, then assemble the sector perimeter with consistent units.
Section 9 · Circular measure
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which expression gives sector area when θ is in radians?
What you will learn
- Use A = ½r²θ for a sector with θ in radians
- Find sector arc length and perimeter
- Solve for a missing angle or radius from sector measurements
Derive sector area from its fraction of a full circle
A sector with angle θ is the fraction θ/(2π) of a full circle. Multiplying that fraction by πr² gives A = ½r²θ, provided θ is in radians.
The corresponding curved edge has length s = rθ. These two formulas use the same angle unit and can be combined or rearranged for unknowns.
If a question gives degrees, convert before using either formula. Arc length is measured in units; area is measured in square units.
Perimeter includes two radii and one arc
A sector perimeter is P = 2r + rθ. Include both straight radial sides unless the sector is joined to another shape along one of them.
Check whether the question asks for the curved arc, the area, or the whole perimeter; these are different quantities.
Pause and explain
A sector has radius 5 cm and angle 2 radians. What is its arc length?
Worked example
For r = 6 cm and θ = 2/3 radians, find the sector area, arc length and perimeter.
Show the worked solution
- Area: A = ½ × 6² × 2/3 = 12 cm².
- Arc length: s = 6 × 2/3 = 4 cm.
- Perimeter: P = 6 + 6 + 4 = 16 cm.
Answer Area 12 cm²; arc 4 cm; perimeter 16 cm.
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.
Find the area and perimeter of a sector with r = 9 cm and θ = 4π/9 radians.
Give me a hint
Use A = ½r²θ and add two radii to the arc for the perimeter.
Compare my reasoning
- A = ½ × 81 × 4π/9 = 18π cm².
- The arc is s = 9 × 4π/9 = 4π cm.
- The perimeter is 18 + 4π cm.
Area 18π cm²; perimeter 18 + 4π cm.
Look for these in your work
- Area formula applied
- Arc computed separately
- Two radii included
A sector has radius 4 m and angle π/3 radians. Find its area, arc length and perimeter.
Give me a hint
Keep π exact and report the area and lengths with different units.
Compare my reasoning
- Area = ½ × 16 × π/3 = 8π/3 m².
- Arc = 4π/3 m.
- Perimeter = 8 + 4π/3 m.
Area 8π/3 m²; arc 4π/3 m; perimeter 8 + 4π/3 m.
Look for these in your work
- Exact area found
- Arc length found
- Perimeter includes both radii
A rotating sprinkler waters a sector of radius 0.25 m through 1.8 radians. Find the watered area.
Give me a hint
Use the radius squared in the sector area formula.
Compare my reasoning
- A = ½r²θ.
- A = ½ × 0.25² × 1.8 = 0.05625.
- The area is 0.05625 m².
0.05625 m².
Look for these in your work
- Radian angle used
- Radius squared
- Square-unit answer stated
Common mistakes
- Using degrees in the radian sector formula
- Forgetting the factor ½ in sector area
- Counting only one radius in a sector perimeter
- Mixing area and length units
What can you explain now?
What is the area of a quarter-circle sector of radius r?
Compare with the explanation
πr²/4
A quarter-turn is π/2 radians, so A = ½r²(π/2).
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Make it stick
How do you derive the sector area formula from the fraction of a full circle, and what belongs in the perimeter?
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.