Solve compound shapes with arcs and sectors
Combine arc lengths and sector areas while counting only the exposed boundary of a compound shape.
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
For a compound shape, which edges count toward its perimeter?
What you will learn
- Find area and perimeter for compound circular shapes
- Use annular sectors and combined straight and curved boundaries
- Keep internal joins out of an external perimeter
Split a compound shape into familiar parts
For a shape built from sectors or circles, calculate each area with the appropriate radius and radian angle, then add or subtract according to the diagram.
For an annular sector, subtract the inner sector from the outer: A = ½θ(R² − r²), where R is the outer radius and r is the inner radius.
When finding perimeter, count only exposed edges. An arc or straight edge shared inside the compound shape is not part of its outer boundary.
Keep each arc's radius and angle attached
The outer and inner arcs of an annular sector have lengths Rθ and rθ. Add both exposed arcs and any exposed radial sides.
If a straight side is joined to another part of the shape, omit that side from the outside perimeter. Mark each boundary piece before adding lengths.
Pause and explain
Which radius is used for the inner arc of an annular sector?
Worked example
An annular sector has outer radius 8 cm, inner radius 5 cm and angle π/3 radians. Find its area and outer perimeter.
Show the worked solution
- Area = ½ × π/3 × (8² − 5²) = 13π/2 cm².
- The outer arc is 8π/3 cm and inner arc is 5π/3 cm.
- The two exposed radial edges are each 8 − 5 = 3 cm, so the perimeter is 6 + 13π/3 cm.
Answer Area 13π/2 cm²; perimeter 6 + 13π/3 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 area and perimeter of an annular sector with R = 10 cm, r = 6 cm and θ = π/4 radians.
Give me a hint
Subtract sector areas; add both arcs and the two exposed radial sides.
Compare my reasoning
- Area = ½ × π/4 × (100 − 36) = 8π cm².
- The arcs total 10π/4 + 6π/4 = 4π cm.
- The radial edges total 2(10 − 6) = 8 cm, so perimeter is 8 + 4π cm.
Area 8π cm²; perimeter 8 + 4π cm.
Look for these in your work
- Area difference calculated
- Both arcs included
- Exposed radial edges counted
A shape is a 12 cm by 6 cm rectangle joined along its 6 cm side to a semicircle of radius 3 cm. Find the total area and outer perimeter.
Give me a hint
The shared diameter is internal; only the semicircle arc and exposed rectangle edges count outside.
Compare my reasoning
- Rectangle area is 12 × 6 = 72 cm²; semicircle area is ½π(3²) = 9π/2 cm².
- The exposed rectangle edges are 12 + 12 + 6 = 30 cm.
- The semicircle arc is πr = 3π cm; perimeter is 30 + 3π cm.
Area 72 + 9π/2 cm²; perimeter 30 + 3π cm.
Look for these in your work
- Component areas added
- Shared side excluded
- Semicircle arc included
A circular track sector has radius 40 m and angle 0.75 radians. Find the arc distance and sector area for planning barriers and surface covering.
Give me a hint
Use s = rθ for barrier length along the curve and A = ½r²θ for surface area.
Compare my reasoning
- Arc distance is 40 × 0.75 = 30 m.
- Sector area is ½ × 40² × 0.75 = 600 m².
- Use metres for the barrier and square metres for covering.
Arc 30 m; area 600 m².
Look for these in your work
- Arc and area kept distinct
- Correct units stated
- Context matched to quantity
Common mistakes
- Adding the inner sector area instead of subtracting it
- Using the same radius for both annular arcs
- Including a shared internal edge in the outer perimeter
- Combining lengths and areas in one total
What can you explain now?
What area formula gives an annular sector with radii R and r and angle θ?
Compare with the explanation
½θ(R² − r²)
Subtract the inner sector area from the outer sector area.
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Make it stick
For a compound circular shape, how do you calculate the area and decide which edges belong to 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.